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D 

SHELDONS' TWO-BOOK SERIES. 



SHELDONS' 



Elementary Arithmetic, 



ORAL AND WRITTEN EXERCISES. 



,*.' OF CO '...-, 

SEP 6 '1W6 '^ I 

Of WASHIN'J, 



(JHELDON & CO.MPANY, 

NEW YORK AND CHICAGO. 

18 86. 



SHELDONS' TWO-BOOK SERIES. 



ISt. 

SHELDONS' ELEMENTARY ARITHMETIC 

2d. 
SHELDONS' COMPLETE ARITHMETIC. 



Copyright, 18S6, by Sheldon &■ Co. 



Electrotypbd by Smith A McDougal, 82 Beekmen St., N. Y. 






PUBLISHERS' PREFACE. 



SHELDON'S ELEMENTARY ARITHMETIC is the first of a 
two-book series published to meet the demands of progressive 
teachers. 

The teachers of primary arithmetic have recently made such rapid 
and extensive advances in the methods employed as to create, by the 
variety and novelty of many of these methods, a sense of uncertainty in 
the minds of the teachers themselves. From so much that is novel, it 
has been the aim of the publishers of this series to select that which is 
acknowledged to be a substantial improvement, and to so arrange the 
matter in two books, that the more progressive work in the lower 
primary grades may be quite in harmony with the more conservative 
work in the higher grades. 

In the preparation of this Elementary Arithmetic, special attention 
has been paid to the laws for the development of the faculties of the 
mind. The powers of observation have been encouraged, by the use 
of objects and pictures, and by reference in the problems to the arith- 
metic of nature, as shown in some of the smaller, yet common forms of 
plant and animal life. The combinations of numbers have been im- 
pressed upon the memory by constant, yet as far as possible, unconscious 
repetition. The judgment of the pupil has been called into exercise, not 
only by the methods of comparison, but by the contrasts afforded by 
blending with the otherwise monotonous work of the elementary rules, 
the simplest exercises under the next higher rules. By the problems 
under the first four rules, the reason has been prepared for the greater 



iv PREFACE. 

exercise required of it in the analysis of problems in the section on 
fractions. 

The methods employed are those which, beginning with lessons on 
objects, or their representative pictures, advance by the usual induc- 
tive processes to a distinct apprehension of the rules and definitions 
reached. 

The grading of the work, the unusually large number of problems, 
oral and written, and the many other useful and varied exercises and 
drill tables, will, it is thought, be highly appreciated by the teacher and 
of great benefit to the pupil. 

The book is an elementary arithmetic, and yet by the addition at 
the end of the book of certain useful business forms and processes, it 
has been made complete for the pupils that leave school at an early 
age. 

In Section IV and the following sections, the definitions and rules 
have purposely been made to correspond with those of the Complete 
Arithmetic, in order that pupils may not be required to learn new defini- 
tions and rules when studying the higher book. The examples and 
problems are of a simpler character, and adapted to the minds of younger 
children. The completion of the Elementary Arithmetic thus becomes 
a thorough preparation for the more difficult work in the Complete. The 
publishers believe, that in these two books, containing so great a number 
of well-graded examples and problems, teachers will find a series of 
thoroughly practical and progressive arithmetics. 

SHELDON & COMPANY. 

New York, Sept. 1, 1886. 



CONTENTS 



SECTION I, 

ONE TO TEN. 

PAGE. 



One, Two, Three, ..... 1 

Four, . . 4 

Five, . . . . 8 

Six, ......... 12 



Seven, 16 

Eight, 18 

Nine, 22 

Roman Notation, 28 



SECTION II. 

TEN TO TWENTY. 



Notation, Numeration, ... 29 

Ten, 32 

Eleven, 37 

Twelve, 41 

Thirteen, 46 

Fourteen, 50 

Fifteen, 55 



Sixteen, 60 

Seventeen, (yQ 

Eighteen, 69 

Nineteen, 75 

Twenty, 78 

Measurement, 83 

Roman Notation, ..... 86 



SECTION III, 

TWENTY TO FORTY. 



Notation, Numeration, . 
Addition, 20 to 30, . 
Subtraction, 20 to 30, . 
Multiplication, 20 to 30, 
Division, 20 to 30, . . 



. . 87 


Addition, 20 to 40, . . . 


. 104 


. . 90 


Subtraction, 20 to 40, . . 


. 110 


. . 96 


Multiplication, 20 to 40, 


. 113 


. . 100 


Division, 20 to. 40, . . 


. 113 


. . 100 


Measurement, .... 


. 120 



VI 



CONTENTS. 



SECTION IV. 

DEFINITIONS AND RULES. 



PAGE. 

Definitions, 122 

Notation, Numeration, . . 123 

Roman Notation, 126 

Addition, 128 



PAGE. 

Subtraction, 136 

Multiplication, 142 

Division, 149 

Miscellaneous Problems, . . 157 



SECTION V. 

COMMON FRACTIONS AND DECIMAL FRACTIONS. 

Notation, Numeration, . . 

Addition, 

Subtraction, 

Multiplication, Principles, 
Division, Principles, . . 



160 


Notation, Numeration, . . 


. 183 


164 


Principles, 


. 186 


167 


Reduction, 


. 187 


171 


Addition, Subtraction, . . 


. 188 


177 


Multiplication, Division, . 


. 190 



SECTION VI. 

UNITED STATES MONEY. 



Notation, 194 

Reduction, 195 

Addition, Subtraction, . . . 196 

Multiplication, 196 

Division, 197 



Fractional Parts, 198 

Business Forms, 200 

Interest, 203 

Profit and Loss, 207 

Measurement, 208 




SECTION I. 



ADDITION. 

1. 1. How many little girls in the 
picture are talking ? 

One and one are how many ? 

One and one are ttvo. 

/ + / = 2. 

How many men in the picture 
are riding on bicycles ? 



Jj,. One and one and one are how 
many ? 
One and one and one are three. 

/+;/ + / = 3.' 

5. How many little girls do you see 
picking flowers ? How many other 
girls ? Two and one are how many ? 

Two and one are three. 

2 + 4=3. 

6. How many parrots do you see on 
the upper branch of the tree ? How 
many do you see on the lower branch ? 
One and two are how many ? 

One and two are three. 

4 + 2 = 3. 

i 



2 ONE, TWO, THREE. 

2. Show with counters that: 

i. 1 + 1 = 2. 9. 3' + 1 = -8. 3. 1 + 2 = 3. 

3. Copy and find the sum : 

4 4 2 

4_ 2_ 4_ 

2 3' 

SUBTRACTION. 

4. 1. If one of the parrots in the picture should fly away, 
how many parrots would be left ? 

2. Three less one are equal to how many ? 

Three less one are equal to two. 

3-4 = 2. • 

3. If the two little girls that are talking should walk away, 
how many of the three little girls would remain ? 

4- Three less two equals how many ? 

Three less two equals one. 

3 - 2 = 4. 

5. Of the two little girls that are talking, one has sat down. 
How many of the two little girls remain standing ? 

6. Two less one are how many ? 

7. Two less two are how many ? 

8. Three less three are how many ? 

5. Shotv with counters that: 

l. 2 — I = 1, 3. 3 — 1 = 2. 5. 3 — 3 = 0. 

*. 2 — 2 =: 0. 43-2 = 1, o. 3 - = 3. 



ONE, TWO, THREE. 
6. Copy and find the differences : 



2 


3 ■ 


5 


/ 


/ 


# 


4 


ORAL PROBLEMS. 





7. 1. Two little girls and one little girl are how many 
little girls ? 

2. Two things and one thing are how many things ? 

3. Two units and one unit are how many units ? 
J^. Two and one are how many ? * 

5. If the little girl picks three flowers and gives two away, 
how many will she then have ? 

6. Carrie had three cents and spent one. How many cents 
did she have left ? 

7. If Emma picks a flower for Jennie, one for Edith, and 
one for herself, how many flowers will she pick ? 

8. Three one's are how many ? Three times one = ? 

9. If the three parrots are given to the three little girls, 
how many parrots will there be for each little girl ? 

10. Three divided by three give how many ? 

11. Two divided by two give how many ? 

12. How many one's are there in three ? 

13. How many units are there in three ? 

1J{,. If you divide a cracker equally between two parrots, 
how much will each parrot receive ? One half. 

15. If you divide a cracker equally among three parrots, 
how much will each parrot receive ? One third. 

16. If you divide two crackers equally between two parrots, 
how many will each parrot receive ? 

17. If you divide three crackers equally among three 
parrots, how many will each parrot receive ? 

18. One half of two is how many ? One third of three = ? 




%>. 




Mw ~zgr"- 



4. Four. 4. 

ADDITION. 

8. i. Count the boats in the picture. 
The houses. The horses. 

2. Three houses and one house are how 
many houses ? 

3. Three and one are how many ? 

Three and one are four. 

3 + / = £. 

J/,. How many horses in the field are 
running ? 

5. How many horses are standing near 
the fence ? 

6. Two and two are how many ? 

Two and two are four. 

2 + 2 = A. 

7. The boy climbs over the fence and 
back four times and each time takes a 
melon. How many melons does he take ? 

8. Four one's are how many ? 

1+1+1+1=? 

9. Show with counters that: 

j. 1 + 3 = 4. 3. 3 + 1 = 4 

*. 2 + 2 =: 4. 4. 4 + = 4. 

4 



FOUR. 5 

10. Copy and find the sums : 

13 2 4 

0_ 4_ 2 3 

4 2 4 ' 

4 2 4 4 

§_\£_± : & 

SUBTRACTION. 

11. 1. If the boy in the picture eats one of the four melons, 
how many melons will remain ? 

2. One melon from four melons leaves how many melons ? 
S. One unit from four units leaves how many units ? 

Four less one equals three. 

4 - 4 =3. 

Jj,. If two of the sail-boats sail away, how many will be left ? 

5. Two from four leave how many ? 

Four less two equals two. 

4-2 = 2. 

6. If the three houses standing together should burn down, 
how many of the houses in the picture would remain standing ? 

12. Show with counters that: 

j, 4 _ I = j. s. 4 — 3 = 1. *r 4 — — 4. 

». 4 — 2 = 2. 4 4 — 4 = 0. o. 2 — = 2. 



6 FOUR. 

13. Copy and find the differences \ 



A 


A 


A 


A 


3 


2 


/ 





2 


2 


3 


3 


4 


2 


/ 


2 



MULTIPLICATION AND DIVISION. 

14. 1. How many of the horses are standing ? How many 
are running ? Two two's are how many ? 

2. If the boy in the picture catches the two horses that are 
standing and then the two horses that are running, how many 
horses will he catch ? 

3. Two times two counters are how many counters ? 

Tivo times two are four. 

2 x 2 = 4. 

Jf. If the boy carries a melon into each of the four houses, 
how many melons will he carry ? 

5. Four times one counter are how many counters ? 

Four times one are four. 

6. How many wagons can be drawn by four horses, if it 
takes two horses to draw one wagon ? 

7. Four counters divided by two are how many counters ? 

Fotir divided by tivo equals two. 

4 - B = £. 



FOUR. 



ORAL PROBLEMS. 



15. 1. If the four sail-boats are owned by the four sailors 
living in the houses, how many sail-boats are there for each 
sailor ? 

2. Four divided by four are how many ? 

3. How many one's are there in four ? How many two's ? 
How many four's ? 

Jj,. If four horses together draw a wagon load of stones, 
what part of the load does each horse draw ? One fourth. 

5. If the four horses draw four different carriages, how 
many carriages will there be for each horse ? 

6. One fourth of four is how many ? 

7. The nearest house in the picture has four front win- 
dows, half of them on either side of the door. One half of 
four is how many ? 

8. There are four weeks in a month. How many weeks 
are there in half a month ? 

9. If an apple is cut into two equal parts, what is the name 
of each part ? 

10. How many halves are there in any one thing ? 

11. If half an apple is cut into two equal parts, what is 
the name of each part ? One quarter. 

12. How many quarters, or fourths, are there in any one 
thing ? 

13. Eddie has two cents, George has two times, or twice, 
as many. How many cents has George ? 

llj,. A square has four equal sides. If each side is one foot 
long, how far is it around the square ? 

15. A triangle has three sides. If the three sides are equal 
and each one foot long, how long are the three sides together ? 

16. If Mary has four cents and spends three, how many 
cents will she then have ? 

17. Katie has four dollars. Frank has one dollar. How 
many more dollars has Katie than Frank ? 




5. Five. 5. 




ADDITION. 



16. i. How mdny boys in the pic- 
ture are coasting ? 

2. How many are lying down on 
their sleds ? How many are sitting up ? 

3. Four and one are how many ? 

Four and one are five. 



4 + 4 



5. 



Jf. How many gold fish are in the 
globe ? How many are just outside 
of it? 

5. Three and two are how many ? 

Three and hvo are five. 

3 + 2 = 5. 

6. Two and three are how many ? 

7. One and four are how many ? 

8. How many squirrels are there in 
the picture ? 

9. How many one's are there in five ? 

10. How many units are there in 
five ? 

17. Shotv with counters that : 

il+4 = 5. 3. 3 + 2 = 5. 



2. 2 + 3 = 5. 



4. 4 + 1 



5. 



8 



FIVE. 
18. Copy and find the sums : 



4 


2 


3 


A 


-L 


3 


2 


4 


3 


2 


4 


3 





2 


2 


4 


2 


4 


2 


4 



SUBTRACTION. 

19. 1. In the glass globe there were five gold fish. After 
two were taken out, how many fish remained in the globe ? 

2. Two fish from five fish leave how many fish ? 
' 3. Two from five leave how many ? 

Five less two equals three. 

5-2 = 3. 

Jj,. One of the five coasters leads and the others follow. 
How many coasters are following the leader ? 

5. One from five leaves how many ? 

6. If three of the five squirrels are frightened away, how 
many will remain ? 

7. Three from five leave how many ? 

Five less three equals two. 

5-3 = 2. 

8. After a little girl picked and ate four of the five berries, 
how many were left ? 5 — 4 = ? 

9. Tommie took the other three fish from the glass globe. 
How many were left then ? 

20. Show with counters that: 

i. 5 — 5 = 0. 3. 5 — 3 = 2 5. 5-1 = 4. 

9% 5 — 4 = 1. 4. 5 - 2 = 3. 6. 5 — = 5. 



10 FIVE. 

21. Write from dictation and find the differences i 

1. 2. 

5 4 

_0 JL 

22. Copy and complete: 

3 + 2 = 
5-3 = 



3. 4. 

5 3 
2 3 


5. 6. 

5. 2 
4 • 2 


uplete : 

4-3 = 


5 -0 = 


2*2 = 


4+'J = 


4 + 2 = 


4 + 4 = 


ORAL PROBLEMS. 





23. i. If four marbles are divided equally between two 
boys, how many marbles will each boy get ? 

2. Edith had two dollars in bank and Harry, three dollars. 
How many dollars in bank did both have ? 

3. Frank bought two tops and found two. How many 
did he then have ? 

Jf. Mary had five peaches and ate two. How many peaches 
had she left ? 

5. On a tree sat four birds. Two flew away. How many 
still sat on the tree ? 

6. Kittie has five cakes. To how many little girls can 
she give one cake each ? 

7. Charlie had three oranges and Ned, one orange. How 
many oranges did the two boys have ? 

8. Mabel earned five cents. After spending three cents, 
how many did she have left? 

9. Howard's skates cost three dollars. His sled cost one 
third as much. How much did his sled cost ? How much 
did sled and skates together cost ? 

10. Roy was four years old. His little brother was only 
half as old. How old was the little brother ? 



FIVE. 11 

11. Frank had two marbles. Mat had one more than 
Frank. How many marbles did both have ? 

12. Mr. Bates walked five miles from home. He walked 
back two miles and rode the rest of the way. How far did 
he ride ? 

IS. If two farmers have each two ponies, how many ponies 
altogether have they ? 

ljf. Edith buys two postage stamps at two cents each. 
How much change should she receive, if she pays for them 
with a nickel, or five-cent piece ? 

15. Mr. Henry bought his little boy a hat for two dollars 
and paid for it with a five-dollar bill. How much money was 
coming to him ? 

16. Eobbie bought a pair of pigeons for four dimes. As 
two make a pair, what was the cost of one pigeon ? 

17. How many gloves are there in two pairs of gloves ? 

18. From a dish containing four oranges, take two. Take 
two more. How many oranges will remain ? 

19. How many two's in four ? 

20. If there are four apples in a dish and you eat one half 
of them, how many do you eat ? 

21. One half of four is how many ? One fourth, or quar- 
ter, of four ? How many two's in four ? How many one's ? 

22. If five dollars were divided equally among five boys, 
how many dollars would each boy receive ? 

23. Three dollars plus, or and, two dollars are how many 
dollars ? 

Three dollars plus ttvo dollars are five dollars. 

$3 + $2 = $5. 

24* Five cents minus, or less, two cents, are how many 
cents ? 

Five cents minus two cents are three cents. 

5* 2* = St 




6. Six. 6. 



ADDITION. 

24. 1. In the picture, how many 
tents do you see beyond the flag ? 

2. How many other tents are there ? 
Five and one are how many ? 

Five and one are six. 

5 + / = 6. 

S. How many soldiers are standing 
ready to march ? 

Jf. How many soldiers are talking ? 
Four and two are how many ? 

5. Count the drums. Three and 
three are how many ? Two three's are 
how many ? 

6. Two and four are how many ? 

7. One and five are how many ? 

8. Two plus two plus two are equal 
to how many ? Three two's = ? 

Two plus two plus two equals six. 

2 + 2 + 2 = 6. 

12 



SIX. 13 

25. Show with counters that: 

i. 5 + 1 = 6. 3. 3 + 3 = 6. s. 1 + 5 = 6. 

*. 4 + 2 = 6. 4.2 + 4 = 6. 6.0 + 6 = 6. 

26. Cop?/ att^ find the sums : 

5 U ' S3 $2 $4 
4 2 3 J, 5 







4 2 2 3 


5 


3 2 4 2 





2 4 3 4 


4 


27. Write from dictation and complete: 




l. 3 + 3 = 3.5 + 1= 5.2x2 = 


7. 4-^4 = 


». 4 + 2= 4.6 + 0= 6.4^2 = 


8. 4x1 = 


SUBTRACTION. 





28. 1. If the tent standing by itself is moved away, how 
many tents will remain ? 

2. One from six leaves how many ? 

Six minus one equals five. 

6-4=5. 

3. When the two soldiers that are talking walk away, how 
many will be left ? 

Jj,. Two from six leave how many ? 

5. If a drummer boy should carry away three of the drums, 
how many would be left ? 

6. Three from six leave how many ? 



14 SIX. 

7. Six dollars minus four dollars equals how many dollars ? 
Six dollars minus four dollars equals two dollars. 

$0 — $J^ — $2. 

29. Show with counters that: 
i. 6 _1 = 5. 

s. 6-2 = 4. 

30. Copy and fin 

6 6 

5 J, 

3 / 



MULTIPLICATION AND DIVISION. 

31. 1. How many skaters are there in each of the two 
groups ? Two three's are how many ? 

2. Two times three counters are how many counters ? 

Tioo times three are six. 

> x 3 = 6. 

3. One half of six is how many ? 

Jf. Three times two counters are how many counters ? 
5. In the picture are six cups divided into three 
How many cups are there in each set ? 



s. 6—3 = 


3. 




5. 


6—5 = 1. 


4. 6—4 = 


:8. 




6. 


6-6 = 0. 


Hie differ 


enees . 








6 




6 




6 


3 




4 




5 


6 




3 




5 


6 




2 




I 



SIX. 15 

Six divided by three equals two. 

6 -*- 3 = 2. 

6. One third of six is how many ? 

7. Six divided by two are how many ? 

8. How many skaters are there in the picture ? How many 
tents ? How many soldiers ? How many drums ? How 
many cups ? 

ORAL PROBLEMS. 

32. _Z. If a lead pencil costs three cents, what will two 
lead pencils cost ? 

2. Edgar earns six dollars a week. Charles earns half as 
much. How much does Charles earn ? 

3. What will be the cost of three bananas, if each one costs 
two cents ? 

^. A fly has six legs. It has only one third as many wings. 
How many wings has a fly ? 

5. A bee has twice as many wings as a fly. How many 
wings has a bee ? 

6. If a bee loses two of its six legs, how many legs will it 
have left ? 

7. Three skaters have how many skates ? 

8. Three two's are how many ? Two three's ? 

9. If I pay three dollars for shoes, two dollars for a hat, 
and one dollar for a pair of gloves, how much do I spend ? 

10. If the shoes and hat and gloves cost six dollars, will 
five dollars pay for them ? How much more will be needed ? 

11. If there are three feet in a yard, how many feet are 
there in two yards ? 

12. Two times three feet are how many feet ? 

13. If each side of a triangle is two yards long, how long 
will the three sides be ? 

14. Henry can lift six pounds. Eddie can lift only one 
third as much. How much can Eddie lift ? 





7. Seven. 7. 

ADDITION. 

33. 1. How many bees are there at the 
right of the hives ? How many in front ? 

2. Six bees and one bee are how many 
bees ? One and six ? 

Six and one are seven. 

6 + 4 = 7. 

3. Count the eggs in the nest. How 
many eggs are there just outside of the 
nest ? 

4- Five and two are how many ? Two 
and five ? 

5. How many plums do you see in the 
picture ? 

6. Four and three are how many ? 
Three and four ? 

34. Show with counters that: 

i. 6 + 1 = 7. * 4 + 3 = 7. 

». 5 + 2 = 7. 4. 2 + 5 = 7. 

35. Copy and complete: 

2+4-5= 2*2-3= 

16 





SEVEN. 






36. Copy and add: 






3 A 


5 


6 


2 


2 4 


4 





3 


4 2 


4 

SUBTRACTION. 


4 


2 



17 



37. 1- If two of the seven bees in the picture enter the 
bee hives, how many will still be flying outside ? 

2. Two from seven leave how many ? Five from seven ? 

3 . If Ned eats three of the seven plums, how many will 
remain ? 

Jf. Seven cents minus four cents are how many cents ? 

Seven cents minus four cents equals three cents. 

7* - P = 3t 

5. Matthew earned six dollars a week. He paid three dol- 
lars for board. How much did he save ? 

6. In a piece of ribbon there were seven yards. After two 
yards had been cut off, how much remained ? 

38. Shotv with counters that: 

i. 7-1 = 6. s. 7-3 = 4. 5. 7-5 = 2. 7. 7-7 = 0. 
2. 7—2 = 5. 4. 7-4 = 3. 6. 7 — 6 = 1. s. 5-5 = 0. 

39. What is the value of: 

1.7-2? 5.7-6? 5.7—1? 7.7-0? 

2. 7 — 4? 4. 7 — 3? 6. 7—5? s. 7 — 7? ; 

40. Write front dictation and find the differences : 

1. 2. 3. 4. 5. o. 

7 7 7 7 7 7 

12 3 4 5 6 





8. Eight 8. 

ADDITION. 

41. 1. How many peaches do you 
see in the picture ? 

2. Seven and one are how many ? 

Seven and one are eight. 

7+4 = 8. 

3. How many ducks do you see in 
the water ? How many on the land ? 

Jf. Six and two are how many ? Two 
and six ? 

5. How many large leaves can you 
count above the peaches in the picture ? 

6. Three and five are how many? 
> Five and three ? 

7. Count the birds in the picture. 
Two four's are how many ? 

42. Show with counters that: 



i. 7 + 1 = 8. 

2 . 6 + 2 = 8. 

3 . 5 + 3 = 8. 

4 . 4 + 4 = 8. 



5 . 3 + 5 = 8. 

6 . 2 + 6 = 8. 

7. 1 + 7 = 8. 

8 . + 8 = 8. 



18 



EIGHT. 19 

43. What is the value of: 



l. 4+2 + 2? 


3. 4 + 4 


? 5. 


8 + 0? 


7. 1 + 3 + 2? 


s. 3 + 1 + 4? 


4. 7 + 1 


? 6. 


6 + 2? 


s. 5 + 1 + 2? 


44. Copy and add: 








7$ 


m 


£Y 


M 


/* 


4 


6 


3 


4_ 


7 


t 


t 


t 






2 


3 


4 


4 


2 


3 


4 





I 





2 


/ 


2 





5 


4 


2 


2 


3 


4 




SUBTRACTION. 





45. 2. If the four birds at the right fly away, how many 
birds will remain on the branch of the tree ? 

2. Four from eight leave how many ? 

3. If the wind blows three of the eight large leaves off the 
tree, how many leaves will be left ? 

J/,. Three from eight leave how many ? Five from eight ? 

5. The two ducks on the land waddle away. How many 
ducks remain ? 

6. Two from eight leave how many ? Six from eight ? 

7. A boy ate one of the peaches. How many were left ? 

8. One from eight leaves how many ? Seven from eight ? 

46. Shoiv with counters that: 

i. 8-1 = 7. 3. 8-3 = 5. 5- 8 — 5 = 3. 7. 8 — 7 = 1. 

9 . 8-2 = 6. 4, 8-4 == 4. <?• 8-6 = 2.U. 8-8 ~=. 0. 



20 EIGHT. 

4*7.. Copy and complete: 

6+3+2= 8-2 = 3*2-4= 
5 + 3-4 = 8-6 = 7-/i+2 = 
6+2+3= 8-5= 2x3-5= 

48. Write from dictation and find the differences : 

1. 2. 3. 4. 5. 6. 7. 8. 

78878668 
J,_3J>J3_4_2_4_2 

MULTIPLICATION AND DIVISION. 

49. 1. Two times four counters are how many counters ? 

Two times four are eight. 

2 X 4 = 8. 

2. Four times two counters are how many counters ? 
•?. One half of eight counters is how many counters ? 
Jf. Eight divided by two are how many ? 

Eight divided by two equals four. 

8 +2=4- 

5. Eight divided by four are how many ? 

Eight divided lojfour equals tivo. 

8 + 4 = 2. 

6. One fourth of eight counters is how many counters ? 

7 . In the picture, how many cherries do you see ? How 
many appear to grow in the same place ? 

8. Four two's make how many ? How many two's in eight ? 

9. How many birds are there in the picture ? How many 
large leaves ? How many peaches ? How many ducks ? 

10. In one eight, how many units are there ? 



EIGHT. 21 

ORAL PROBLEMS. 

50. 1. Anna had seven needles and lost two. How many 
needles did she then have ? 

2. A little girl had five needles and found two. How 
many needles did she then have ? 

3. Eddie had six books. After he gave half of them to 
his sister, how many did he have ? 

Jf. Mrs. Barnes bought four canary birds. How many 
cages were needed if she put two birds into each cage ? 

5. How many men are working on a farm, if two are 
working in one field and three in another ? 

6. Helen had six paper dolls. She gave two of them to 
Maggie. How many did she then have ? 

7. If there are two trees in front of each house, how many 
trees are there in front of four houses ? 

8. Richard has four pigeons. Frank has three more than 
Richard. How many has Frank ? 

9. Mr. Green had eight dollars. He spent two dollars and 
lost three. How many dollars did he then have ? 

10. Fred had six cents. He spent two cents and found 
four. How many cents did he then have ? 

11. Howard spent two dollars for fire-crackers, one dollar 
for torpedoes, and four dollars for other fireworks. How 
much money did he spend ? 

12. A lily has six petals, or leaves, and a pansy has five. 
How many more petals has the lily ? 

13. On a shelf are eight story books. One fourth of them 
belong to Fannie. How many of the books are Fannie's ? 

14- Each leaf of a book has two pages. How many pages 
have four leaves ? 

15. One horse has four legs. How many legs have two 
horses ? 

16. Frank had seven cents. He bought three peaches at 
two cents each. How many cents did he have left ? 





(j. Nine. (j. 

ADDITION. 

51. 1. How many pears can yon 
connt in the picture ? 

2. Eight and one are how many ? 

Eight and one are nine. 

8 + / = (j. 

3. Seven kittens and two kittens 
are how many kittens ? 

4. Two and seven are how many ? 
One and eight ? 

5. How many children's faces do 
you see in the large circle in the 
picture ? 

6. How many faces are there in 
the small circle ? 

7. Six and three are how many ? 
Three and six ? 

8. How many pansies can you count 
in the corner of the picture ? 

9. Five and four are how many ? 
Pour and five ? 

22 



NINE, 



23 



10. If Fannie spends five cents for candy, two cents for 
cakes, and two cents for licorice, how much does she spend ? 

11. Hattie saved four dollars the first month, three dollars 
the second month, and two • dollars the third month. How 
much did she save in the three months ? 

12. Three and three and three equals how many ? 

Three plus three plus three equals nine. i 

3 + s + 3 = (j. 

13. Three three's are how many ? 

14- Mr. Arnold had four children and his brother had 
five. How many children did both have ? 

15. In one house there are two children. In another 
house there are seven. How many children are there in the 
two houses ? 

16. Mattie had four two-cent pieces and a penny. How 
much money did she have ? 

17. Lucy is six years old. If her sister Emma is three 
years older, how old is Emma ? 

52. Show with counters that: 

i. 8 + 1 = 9. 4. 5 + 4 = 9. 

2. 7 + 2 = 9. s. 4 + 5 = 9. 

s. 6 + 3 = 9. <?. 3 + 6 = 9. 

53. Write from dictation and add : 
l. 2. 3. 4. 

$3 U $5 H 

3 2 10 

3 3 3 3 



6. 

2 
1 
4 

2 



7. 

2 
4 
2 
1 



8. 

5 


1 

2 



9. 

4 
1 

4 



7. 


2 + 7 = 


:9 


8. 


1 + 8 = 


:9. 


9. 


+ 9 = 

s. 

2<? 

4 

3 

lO. 

3 
2 
1 
3 


9. 



24 



NINE. 



54. Announce sums at sight : 

Read answers : 1. From left to right ; 2. From right to left ; 3. From 
top to bottom ; 4. From bottom to top ; 5. In any order suggested by 
the teacher. 



1st. 2d. 3d. 4th 5th. 6th. 7th. 8th. 9th. 
345213463 
423421016 



C 



D 



2 


3 


4 


2 


1 


2 


1 


8 


7 


1 


3 


5 





4 


2 





1 


1 


3 


4 


2 


6 


1 


2 


3 


4 


1 


5 


1 


6 


3 


5 


7 





4 


1 


7 


2 


2 


6 


1 


4 


5 


1 


7 


2 


5 


3 





7 


3 





8 






1 

6 


5 
4 


8 19 5 
3 2 

SUBTRACTION. 


6 
2 


3 

'2 


5 
1 



E 



55. 1. If Martha eats one of the nine pears in the 
picture, how many pears will remain ? 

2. One from nine leaves how many ? Eight from nine ? 

3. Jesse frightened the two little kittens at the left and 
they ran away. How many of the nine kittens remained ? 

Jf,. Two from nine leave how many ? Seven from nine ? 

5. In the two circles in the picture there are nine faces. 
There are three faces in the smaller circle. How many are 
there in the larger circle ? 

6. Three from nine leave how many ? Six from nine ? 

7. If some little girls pick four of the nine pansies in the 
picture, how many will remain ? 



NINE. 



25 



8. A dish and a plate together hold eight apples. Three of 
the apples are on the plate. How many are in the dish ? 

9. On a tray are eight rings. Only two of the rings are 
made of gold. How many of them are not made of gold ? 

56. Announce differences at sight : 

Read answers : 1. From left to right ; 2. From right to left ; 3. From 
top to bottom ; 4. From bottom to top ; 5. In any order suggested by 
the teacher. 

5th. 6th. 7th. 8th. 9th. 

7 5 8 4 9 

5 15 19 



B 



1st. 


2d. 


3d. 


4th. 


9 


7 


8 


4 


5 


2 





2 



c 



6 


9 


8 


7 


4 


6 


9 


6 


6 


3 


2 


4 





3 


6 


7 


1 





5 


8 


6 


1 


5 


7 


3 


8 


3 





1 


4 


1 


5 


1 


2 


8 


1 



57. Show with counters that. 



j. 9 — 1 = 8. 

2. 9 — 2 = 7. 

3. 9-3 = 6. 



4. 9 — 4 = 5. 

5. 9 — 5 = 4. 
g. 9 — 6 = 3. 



7. 9-7 = 2. 
s. 9 — 8 = 1. 
9. 9 — 9 = 0. 



58. Copy and complete: 

q - 7 = 8 -*- J, + 5 = 
(j -5 = 5-4 + 8 = 
(j -3 = 2*4-2 = 



x 2 

- 9> 



4 

8 + 

(j - 6 



59. Write from dictation and find the differences t 



2. 



3. 



4. 

6 



4 



26 NINE. 



6. 


7. 


8. 


9^ 


W 


$9 


5 


7 


8 



9. lO. 

9 5 



60. Announce differences at sight; 



Read answers : 1. From left to right ; 2. From right to left ; 3. From 
top to bottom ; 4. From bottom to top ; 5. In any order suggested by 
the teacher. 

1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 

768594637 

7_5_6 4 3_0_1J5_6 

594821789 
3_8_4_2 °J: 37 4 

67293295 
24110262 



MULTIPLICATION AND DIVISION. 



'{. 



61. 1. How many faces are there in the picture ? How 
many kittens ? How many flowers ? How many pears ? 
How many leaves ? 

2. How many nines have you found in the picture ? 

3. In one nine, how many units are there ? 
Jf- Nine units make how many nines ? 

5. Three three's are how many ? 

6. Three times three counters are how many counters ? 

Three times three are nine. 

3 x s = (j. 

7 . Nine counters divided by three are how many counters ? 

Nine divided by three equals three. 

(j + 3 = 3. 



NINE. 27 

8. When postage stamps cost three cents each, what was 
the cost of three stamps ? 

9. If one orange costs three cents, how many can you buy 
for nine cents ? 

10. A man earns nine dollars in three days. How much 
does he earn in one day ? One third of nine is how many ? 

11. A boy received three dollars for one week's work. 
How much should he receive for three weeks' work ? 

ORAL PROBLEMS. 

62. 1. Three plus three are how many ? 3 + 3 = ?. 

2. Three minus three are how many ? 3 — 3 = ? 

3. Three times three are how many ? 3x3 = ? 
4- Three divided by three are how many ? 3-f-3 = ? 

5. Three three's added together make how many ? 

6. Three times three are how many ? 

7. How many threes are there in nine ? 

8. Nine divided by three are how many ? 

9. How many times can you take three from nine ? 

10. 2 + 2=? 2-2=? 2x2=? 2-^2=? 

11. 4 + 4=? 4—4=? 2x4=? 4^4=? 

12. 1 + 1=? 1 — 1 = ? 1x1 = ?' 1-7-1 = ? 

13. What is one half of two ? Of four ? Of six ? Of eight ? 
IJj,. What is one third of three ? Of six ? Of nine ? 

15. What is one fourth of four ? Of eight ? 

16. What is one fifth of five ? 

17. Beginning with naught, add by two's to eight. 

18. Beginning with one, add by two's to nine. 

19. Beginning with naught, add by three's to nine. 

20. Beginning with one, add by three's to seven. 

21. Beginning with two, add by three's to eight. 

22. Beginning with naught, add by four's to eight. 

23. Beginning with one, add by four's to nine. 
24< Subtract by two's from eight. From nine. 

25. Subtract by three's from nine. From eight. From seven. 



28 ROMAN NOTATION. 



ROMAN NOTATION. 

63. In the Roman System of Notation, letters are 
employed to represent numbers. Thus : 

The letter I. stands for 1. 
The letter V. stands for 5. 

64. These two letters are repeated and combined to repre- 
sent different numbers. Thus : 



I. 


II. 


III. 


IV. . 


V. 


VI. 


VII. 


VIII. 


1 


2 


3 


4 


5 


6 


7 


8 



PRINCIPLES. 

65. To repeat a letter repeats its value. 

Thus, II. is 2, and III. is 3. 

The letter I is never used more than three times in any one combina- 
tion. The letter V is never repeated. 

66. When one letter follows another of greater value, 
the sum of their values is denoted ; when it precedes, 
the difference between their values is denoted. 

Thus VI. is 6, and IV. is 4. 

67. Express by figures, or the Arabic Notation: 



1. V. 3. VI. 5. I. 


7. II. 


2. IV. 4. VII. 6. III. 


8. VIII. 


68. Express by the Roman Notation: 




1. 5. 3. 7. 5. 1. 


7. 3. 


2. 6. 4. 8. e. 2. 


s. 4. 



SECTION II. 



NOTATION AND NUMERATION. 

1. How many units are represented by each of the follow- 
ing figures : 

123456789 0. 

2. Write the words for which the above figures stand. 
□□□□□□□□□□ make 1 1 1 1 1 I I ITT1 



Ten Units 



make 



One Ten, 



69. 1. Ten units are represented, not by a single figure, 
but by writing 1 at the left of ; thus, 10. 
2. In the same manner : 



Eleven units, 


or 1 ten 


and 


1 unit, 


are 


written 


11. 


Twelve units, 


" 1 ten 


a 


2 units, 


a 


a 


12. 


Thirteen units, 


" 1 ten 


a 


3 units, 


a 


a 


13. 


Fourteen units, 


" 1 ten 


a 


4 units, 


a 


a 


14. 


Fifteen units, 


" 1 ten 


a 


5 units, 


a 


£( 


15. 


Sixteen units, 


" 1 ten 


a 


6 units, 


i( 


a 


16. 


Seventeen units, 


" 1 ten 


a 


7 units, 


a 


a 


17. 


Eighteen units, 


" 1 ten 


a 


8 units, 


a 


a 


18. 


Nineteen units, 


" 1 ten 


a 


9 units, 


£( 


a 


19. 



70. Show with counters the meaning of: 

10, 11, 12, 13, 14, 15, 16, 17, 18, 19. 

29 



30 



NOTATION AND NUMERATION 



GROUPS OF TEN. 
1. How many tens in each of the following groups ? 



71. i. To represent one ten, or ten units, the figure 1 is 
written at the left of the cipher, ; thus, 10 ; to represent 
two tens, or twenty units, the figure 2 is written at the left 
of the cipher, ; thus, 20. 

2. In the same manner : 



Three tens, 


or thirty 


units, 


are 


written 30 


Four tens, 


" forty 


units, 


a 


40 


Five tens, 


" fifty 


units, 


a 


50 


Six tens, 


" sixty 


units, 


a 


60 


Seven tens, 


" seventy 


units, 


a 


70 


Eight tens, 


" eighty 


units, 


a 


80 


Nine tens, 


" ninety 


units, 


a 


90 



72. Show with counters the meaning of: 

10, 20, 30, 40, 50, 60, 70, 80, 90. 

TENS AND UNITS. 

1. How many units are represented by each of the follow- 
ing numbers ? 

10, 11, 12, 13, 14, 15, 16, 17, 18, 19. 



NOTATION AND NUMERATION 



31 



2. For what does the figure, I, in 10 stand ? In 12 ? In 
15 ? Is it in the first, or the second place ? Does the figure 
in the first, or right-hand place represent tens or units ? 

3. To represent two tens and three units, where should the 
2 be written ? The 3 ? Why ? 

Jj,. To represent four tens and five units, where must the 
4 be written ? The 5 ? Why ? 



73. Numbers composed of tens and units are expressed by 
writing the figure representing the units at the right of the 
figure representing the tens. Thus, 



2 tens and 3 units, or 

3 tens " 4 units, " 



4 tens 

5 tens 

6 tens 

7 tens 



4 units, " 

6 units, " 

3 units, " 

8 units, " 



8 tens " 9 units, " 

9 tens " 9 units, " 



twenty-three, 

thirty-four, 

forty-four, 

fifty-six, 

sixty-three, 

seventy-eight, 

eighty-nine, 

ninety-nine, 



are written 23. 
34. 
44. 
56. 
63. 
78. 
89. 
99. 



74. Show with counters the meaning of the following 
numbers : 



1. 


20, 


23, 


25, 


5. 


60, 


67, 


69j 


2. 


30, 


34, 


36, 


6. 


W, 


75, 


78, 


3, 


40, 


45, 


48, 


7. 


80, 


84, 


86, 


4. 


50, 


52, 


55, 


S. 


90, 


93, 


99. 



75. Write the following numbers: 



1. 


Thirty-nine. 


5. 


Eighty-one. 


9. 


Ninety-four. 


2. 


Sixty-eight. 


6. 


Forty-six. 


10. 


Thirty-seven 


3. 


Ninety-seven. 


7. 


Twenty-two. 


11. 


Fifty. 


4- 


Fifty-three. 


8. 


Nineteen, 


12. 


Sixty-five. 



32 



TEN. 



10. TEN. X. 

ADDITION. 

76. 1. Charles has nine cents. 
If he earns 1 cent more, how many 
cents will he have ? 

2. John is 8 years old. His sis- 
ter, Mary, is 2 years older. How 
old is Mary ? 

3. Carrie saw 7 cows in one field 
and 3 in another. How many cows 
did she see in both fields ? 

Jf. A gentleman gave $6 for a 
pair of shoes and $4 for a hat. 
How many dollars did the shoes 
and hat together cost ? 

5. Fred and Fannie have each 
$5. How many dollars have they 
together ? 

6. Add: 



9 


+ 


1 


= 10 


1 1 1 1 


8 


+ 


2 


= 10 


1 1 1 i 


7 


+ 


3 


= 10 


MM 


6 


+ 


4 


= io 


1 1 1 


5 


+ 


5 


= 10 


MM 



1. 


3. 


s. 


4. 


5. 


1 


2 


3 


4 


5 


9 


8 . 


7 


6 


5 


6. 


7- 


8. 


9. 


10 


6 


7 


8 


9 


10 


4 


3 


2 


1 






7. How many are 2 and 8 ? 7 and 3 ? 3 and 7 ? 

8. If Tom catches 1 fish and his brother, William, catches 
9 fish, how many do they both catch ? 

9. If Katie has 2 walnuts and Martha 8, how many have 
they together ? 

10. Tillie bought an orange for 3 cents and a loaf of bread 
for 7 cents. How much did she give for both ? 




TEN. 33 

11. Nat paid 5 cents car-fare in going on an errand, and 
5 cents, in returning. How much did 
both rides cost him ? 

12. Two fives are how many ? 

77. Write from dictation and add : 

Explanation. — 1. The numbers are written so 
1 ' that the units stand in one column and the tens in 

/ J another. 

2. 4 units and 2 units are 6 units, and 3 units 
&0 are 9 units, and 1 unit are 10 units or 1 ten and 

0^ units. The is written in units' column and the 1 

*3 & ten is carried to the tens' column. 

G) I 3. The 1 ten from units' column and 2 tens are 3 

____ tens, and 3 tens are 6 tens, and 2 tens are 8 tens, 

si r\ and 1 ten are 9 tens. The 9 is written in the tens' 

/ ^ column ; 9 tens and units, or 90, is the sum, 

or answer. 
Note. — In practice, in adding a column, announce the sums only ; 
thus, in units' column, 6, 9, 10 ; in tens' column, 3, 6, 8, 9. Answer, 90. 



2. 


3. 


4. 


5. 


6. 


7. 


16 


25 


12 


17 


40 


32 


31 


13 


23 


30 


25 


13 


12 


21 


30 


21 


12 


34 


31 


40 


24 


31 


21 


11 



WRITTEN PROBLEMS. 

78. 1. Kalph has 68^ and Willie, 12^ ; how many cents 
have both ? 

2. Mr. Flagg is 56 years old. How old will he be in 
14 years ? 

3. In a class are 31 boys and 29 girls. How many pupils 
are there in the class ? 

Jj,. Susie saw 53 carriages in the park on Monday and 37 
carriages on Tuesday. How many carriages did she see in 
the park in the two days ? 



34 



TEN. 



10 - 1 = 9 


10 - 9 = 1 


10 - 2 = 8 


lO- 8 = 2 


10-3 = 7 


10— 7 =3 


10 - 4 = 6 


10-6 = 4 




10 - 5 = 5 





2 from 10 leave how many ? 



10. TEN. X. 

SUBTRACTION. 

79. 1. Hattie had 10 
cents and spent 1 cent for 
candy. How many cents 
did she have left ? 

2. 1 added to 9 makes 
how many ? 1 taken from 
10 leaves how many ? 

3. 9 added to 1 make 
how many ? 10 — 9= ? 

£. 2 and 8 are how many ? 
8 from 10 ? 

5. Horace earned 10 cents by running errands. He lost 
2 cents. How many cents did he then have ? If he had lost 
8 cents, how many cents would have been left ? 

6. 3 and 7 are how many ? 3 from 10 leave how many ? 
7 from 10 ? 

7. Andrew earned $10 a week. He spent $3 for board. 
How much was left ? 

8. 6 from 10 leave how many ? 4 from 10 ? 

9. Of 10 answers to problems, 4 were wrong. How many 
were right ? 

10. Ida bought a bag of salt for 5 cents, how much change 
should she receive out of 10 cents ? 

80. Copy and complete: 

i 10 — 1= s. 10 — 8= n. 10 

2.10 — 9= 4.10 — 2= e. 10- 

81. State differences at sight: 

l. 2. 3. 4. 5. 6. 

10 10 10 10 10 10 
7 9 2 5 4 1 



-3 = 


7. 10- 


-4 = 


-7 = 


8. 10- 


-6 = 


7. 


S. 9. 


10. 


10 


10 10 


10 


3 . 


6 8 


10 



TEN. 35 

82. Write from dictation and subtract : 

l. Explanation. — 1. The less number is written 

Q £Z under the greater, so that units are written under 

/ units and tens under tens. 

A,S 2. 3 units from 5 units leave 2 units. Write 2 

in units' column. 

£) 9} S. 4 tens from 9 tens leave 5 tens. Write 5 in 
tens' column. Answer, 5 tens and 2 units, or 52. 

Note. — In practice think and write only the differences ; thus, 2, 5. 
Answer, 52. 



s. 


3. 


4. 


s. 


89 


76 


58 


27 


27 


35 


46 


14 


6. 


7. 


8. 


9. 


90 


84 


43 


66 


80 


51 


22 


32 



WRITTEN PROBLEMS. 

83. i. Mr. Harvey received $84 for his week's wages. 
His expenses were $62. How much remained each week ? 

2. Mr. Brown is 58 years old. His wife is 14 years 
younger. How old is his wife ? 

3. An acre of land cost $75. It was sold for $60. How 
much money was lost ? 

Jf. If I buy a wagon for $52 and sell it for $79, how much 
do I make ? 

5. A lady paid $87 for her parlor carpet and $35 for her 
dining-room carpet. How much more did the parlor carpet 
cost ? 

6. Mr. Smith walked 68 miles and rode 99 miles. He rode 
how much further than he walked ? 

7. A reader cost 48 cents ; a speller, 24 cents. How much 
more did the reader cost ? 



36 



TEN. 



2x5 = 


io 


10^-2 = 5 


5x2 = 


io 


lO-s-5 = 2 



10. TEN, X. 
MULTIPLICATION AND DIVISION. 

84. 1. Lillie has two 
five cent pieces. How 
much money has she ? 

k Add:' 

l. 5 + 5. 2. 2 + 2 + 2 + 2 + 2. 

3. Two 5's are how many ? Five 2's ? 
4* Two times 5 are how many ? Five times 2 are how 
many ? 

5. Twice 5 are how many ? 

6. If Lillie divides her 10 cents equally between 2 little 
girls, how much will she give to each ? 

7. How many 5's in 10 ? 10 cents divided between 2 girls 
give how many cents to each ? 10 divided by 2 give how 
many ? 

8. One half of 10 is how many ? 



&k&£& 



9. If Eddie divides 10 cents equally among 5 little boys, 
how much will he give to each ? 

10. How many 2's in 10 ? 10 apples divided among 5 boys 
give how many apples to each ? 10 divided by 5 give how 
many ? One fifth of 10 is how many ? 

11. 1 nickel is equal to 5 pennies. To how many pennies 
are 2 nickels equal ? 

12. If there are 5 desks in a row, how many desks are 
there in 2 rows ? 

13. If a yard of muslin costs 10 cents, what will half a 
yard cost ? 

ljf- When milk is 10 cents a quart, what will be the cost 
of a pint, or half of a quart ? 



ELEVEN. 



37 



11. ELEVEN. XL 

ADDITION. 

85. 1. Henry caught 10 trout, 10 + 1 = 11 

and his sister Katie, 1. How many [ I 1 1 I I I I 
trout did they both catch ? 

2. Willie bought 9 marbles and 
found 2. How many marbles did 
he then have ? 

3. Tom sold 8 kites and had 3 
How many had he at first ? 

Add: 



left. 

i 

10 



2 
9 



3. 

3 



4. 

4 

7 



5. 

5 
6 



6. 


7. 


s. 


». 


10. 


6 


7 


8 


9 


10 


5 


4 


3 


2- 


1 



9 + 2= 11 



d 



8 + 3 = 11 



6 + 5= 11 



JL 



B 



B 



U I I I I I 1 l lg 

7 + 4= 11 



5. If farmer Eichards has 7 cows and buys 4 more, how 
many cows will he have ? 

6. Mary pays 6 cents for peaches and 5 cents for bananas. 
How much does she spend ? 

7. Emma saved 7 pennies one week and 4 pennies the 
next week. How many pennies did she save in the two weeks ? 

8. Fannie's bonnet cost $6 and her shoes, $5. What did 
both cost ? 



86. Write from dictation and add : 



1. 


2. 


3. 


4. 


5. 


3 


4 


2 


5 


3 


5 


2 


7 


2 


1 


3 


5 


2 


4 


7 



38 ELEVEN. 

87. Write from dictation and add : 

l. Explanation. — 1. Write the numbers so that 

G) Q units stand in one column, and tens in another. 

2. In the units' column there are 11 units or 
/ Q 1 ten and 1 unit. Write the 1 unit in units' column 

y ^ and carry the 1 ten to tens' column. 

7 & 3. In the tens' column, including the 1 ten car- 

ried from the units' column, there are 8 tens. 
O 4 Write the 8 in tens' column. Answer, 8 tens and 

1 unit, or 81. 



a. 


3. 


4. 


s. 


27 


54 


15 


31 


41 


13 


32 


24 


13 


24 


43 


42 


6. 


7. 


s. 


9. 


12 


20 


18 


10 


33 


15 


30 


5 


40 


24 


22 


23 


14 


32 


11 


43 



WRITTEN PROBLEMS. 

88. i. If a coat costs $39 and a pair of boots $12, what 
will both cost ? 

2. What will be the cost of a carpet at $57 and a rug at $14 ? 

3. A man is now 36 years old. How old will he be in 
15 years ? 

4- What will be the cost of paper for 28 cents, lead pencils 
10 cents, and postal cards 3 cents ? 

5. If there are 37 pins in one cushion and 14 in another, 
how many pins are there in the two cushions ? 

6. On a railroad train there are 23 passengers in the first 
car, 30 in the second car, and 38 in the third car. How 
many passengers are there in the three cars ? 

7. What is the sum of 45, 34, and 12 ? 



ELEVEN. 



39 



11. ELEVEN. XL 
SUBTRACTION. 



11 - 1 = 


10 


11 - 


10= 1 


11 - 2 = 


9 


11 - 


9 = 2 


11 -3 = 


8 


11 - 


8 = 3 


11 -4 = 


7 


11 - 


7 =4 


11 - 5 = 


6 


11 - 


6 = 5 



89. 1. Fred is 9 years 
old. How old will he be 
in 2 years ? 

2. Fred is now 11 years 
old. His sister, Bessie, 
is 2 years younger. How 
old is she ? 

3. 2 years from 11 years 
leave how many ? 2 units from 11 units leave how many ? 
2 from 11 how many ? 9 from 11 ? 

4- If on one plant there are 8 flowers, and on another 3, 
how many flowers are there on both plants ? 

5. From 11 flowers take 3 flowers; how many remain? 
Take 8 ? 

6. Carrie has 11 cents. She spends 4 cents for paper dolls. 
How many cents has she left ? 

7. From 11 take 4. How many remain ? From 11 take 7. 

8. If I have $11 and spend $6, how many dollars shall I 
have ? 6 from 11, how many ? 5 from 11 ? 

90. Copy and complete: 

1. 11— 1= 3. 11 — 2zzz 

2. 11 — 10= 4. 11 — 9 = 

91. State differences at sight: 
l. 2. 3. 

11 11 11 

1 2 3 



6. 

11 

6 



7. 
11 

7 



11 

8 



5. 11—3 = 


7. 


11—4 


6. 11 — 8 = 


8. 


11 — 7 


4. 




5. 


11 




11 


4 




5 


9. 




10. 


11 




11 


9 




10 



40 ELEVEN. 

92. Write from dictation and subtract; 



1. 



Explanation. — 1. In subtraction, write 
the less number under the greater, placing 
units under units and tens under tens. 

2. 6 units taken from 9 units leave 3 
units. Write 3 in units' column. 

3, 1 ten taken from 2 tens leaves 1 ten. 
djk -J A Write 1 in tens' column. The dollar sign, 

iP J O AUS. £ written before the 13 gives the answer, 

$13. 



s6. 



2. 


3. 


4. 


5. 


78 


39 


85 


97 


26 


12 


31 


64 


e. 


7. 


8. 


9. 


64 


46 


25 


86 


53 


24 


14 


55 



WRITTEN PROBLEMS. 

93. i. If I buy a cow for $37 and sell it for $39, how 
many dollars do I make ? 

2. When the price of land is raised from $55 to $68 per 
acre, how much is the increase ? 

3. In one barrel there are 29 gallons of molasses ; in 
another, 23. The first barrel contains how many more gal- 
lons of molasses than the second ? 

4- A man bought a wagon for $69 and sold it for $53. 
How much did he lose ? 

5. A farmer had 95 acres in his farm. He gave 32 acres 
to his son. How many acres had he left ? 

6. If a storekeeper cuts 26 yards from a piece of cloth 
containing 88 yards, how many yards remain ? 

7. Which is the greater 48 or 59 ? How much the greater 
is it ? 



TWELVE. 41 

12. TWELVE. XII. 
ADDITION. 

94. 1. If a coat costs $10 and lO + 2 = 12 

a hat, $2, how much will both 
cost ? $10 and $2 are how many 

dollars ? 10 units and 2 units are 9 + 3 = 12 

how many units ? 10 and 2 are 
how many ? 

2. If 9 yds. of calico make a 
dress and 3 yds. a. shirt, how many 
yards will make a dress and a shirt ? 

3. There are 8 birds in one tree, 
and 4 in another. How many 7 + 5 = 12 
birds are there in the two trees ? i — j — \ — y~y~\ — r~i — 

Jf. How many gold fish are there 
in two globes, when there are 7 in _ Q 

the one and 5 in the other ? 



j. 



i 



8 + 4= 12 






MINIM 













5. I pay $6 for one hat and $6 I I I 1 1 I I, I 
for another. How much do I pay 
for both ? Two 6's are how many ? 

6. Walter walks 4 miles in the first hour, 3 in the second, 
and 5 in the third. He walks how far in the three hours ? 

7. Sadie canned 6 jars of peaches, 4 jars of pears, and 2 
jars of cherries. How many jars of fruit did she can ? 

95# State sums at sight: 

1. 2. 3. 4. 5. 6. 

1 2 3.4 5 6 

11 10 9 8 7 6 



7. 


8. 


9. 


io. 


n. 


is 


2 


2 


3 


4 


6 


7 


5 


9 


1 


2 


2 





5 


1 


6 


4 


4 


3 



42 





TWELVE. 




VriU 


? from dictation and add : 




1. 


2. 


3. 


4. 


29 


40 


45 


25 


30 


26 


34 


30 


23 


16 


11 


26 


5. 


«. 


7. 


*. 


12 


24 


20 


30 


37 


16 


32 


15 


10 


21 


23 


12 


23 


30 


7 


33 



WRITTEN PROBLEMS. 

97. 1. July has 31 days and August has 31 days. How 
many days are there in July and August ? 

2. On a ferry boat were 43 ladies, 36 gentlemen, and 13 
children. How many passengers were there ? 

3. Florence read 34 pages of a book on Monday, 27 pages 
on Tuesday, and 21 pages on Wednesday. How many pages 
did she read in the three days ? 

J/,. On a bridge are 17 cars and 25 wagons. How many 
of both are there ? 

5. In one store a lady spent 33 cents ; in another, 27 cents ; 
in a third, 31 cents. How much did she spend in the three 
stores ? 

6. On the first shelf of a book-case there are 25 books, on 
the second shelf 35 books, and on the third shelf 22 books. 
How many books are there on the three shelves ? 

7. Henry saves 37 cents the first week, 32 cents the second 
week, and 23 cents the third week. How much does he save 
in the three weeks ? 

8. A wagon cost 32 dollars ; a horse, 50 dollars ; and the 
harness, 10 dollars. What was the cost of all ? 

9. Find the sum of 22, 44, and 26. 

10. Find the sum of 16, 40, and 36. 



TWEL VE. 



43 



12. TWELVE. XII. 



12 - 1 = 11 


12 - 11 = 1 


12 - 2 = 10 


12 - 10 = 2 | 


12-3= 9 


12 - 9 = 3 


12 -4 = 8 


12- 8 = 4 


12-5= 7 


12 - 7 = 5 




12 - 


6 = 6 





SUBTRACTION. 

98. 1. 11 eggs and 1 
egg are 12 eggs. 1 from 
12 leaves how many ? 11 
from 12 ? 

2. How many eggs are 
left, if 2 eggs are taken 
from 12 eggs ? 10 from 
12? 

3. Out 4 yds. from a 
dress pattern of 12 yds., 
and how many yards do you leave in the pattern ? 8 from 
12= ? 

4* From a foot, or 12 inches, of brass wire, 5 inches were 
melted. How many inches are left ? 7 from 12 = ? 
5. Of a dozen, or 



12, sheets of paper, 6 
have been sold. How 
many sheets are un- 
sold ? 6 from 12 = ? 
6. On the shore 
were 12 little ducks ; 
4 went into the water. 
How many remained on the shore ? 

99. State differences at sight : 




1. 


2. 


3. 


. 4. 


5. 


6. 


12 


12 


12 


12 


12 


12 


1 


2 


3 


4 


5 


6 


7. 


8. 


9. 


10. 


n. 


12. 


12 


12 


12 


12 


12 


12 


7 


8 


9 


10 


11 


12 



44 



TWEL VE. 



100. Write from dictation and subtract: 



1. 


2. 


3. 


4. 


5. 


99 


87 


54 


93 


48 


33 


65 


43 


61 


12 


6. 


7. 


«. 


9. 


10 


66 


31 


47 


86 


93 


22 


10 


21 


15 


43 



WRITTEN PROBLEMS. 

101. 1. Mr. Baker paid $43 a month, rent. He moved, 
and now pays $12 less per month. How much is his rent ? 

2. Mr. Wilby buys an overcoat for $55. He pays $23 cash. 
How much does he owe ? 

3. Mattie paid 89 cents for fringe. She could have bought 
other fringe for 63 cents. How much more did she pay ? 

J±. A farmer brought 85 quarts of berries to market. He 
sold 73 quarts. How many quarts did he have left ? 

5. One milk can contained 36 quarts of milk, and another, 
32 quarts. How much more did the first can contain ? 



12. TWELVE. XII. 

MULTIPLICATION AND DIVISION. 



2x6=12 
6 x 2 = 12 



102. 1. A lily has 6 
petals, or leaves. How 
many petals have 2 lilies ? 
Two 6's are how many ? 
Twice 6 are how many ? 

2. 2 lilies have 12 petals. 
How many petals has 1 lily ? 

How many 6's are there in 12 ? 12 divided by 2 
half of 12 is how many ? 



3x4=12 
4 x 3 = 12 



12 -f- 2 = 6 
12 h-6 = 2 



12 r3 = 4 
12 -f-4 = 3 



? One 



TWELVE. 



45 




5. 



3. One butterfly has 4 wings. How 
many wings have 3 butterflies ? Three 
times 4 wings are how many wings ? 
Jf. 3 butterflies have 12 wings. How many 4's 
are there in 12 ? 12 divided by 3 = ? One third 
of 12 = ? One fourth of 12 = ? 
.6 + 6 = ? 44-4 + 4= ? 

3 + 3 + 3 + 3=? 2 + 2 + 2 + 2 + 2 + 2=? 

6. Two 6's are how many ? Six 2's are how many ? 

7 . Four 3's are how many ? Three 4's are how many ? 

£. Two times 6 are how many ? Six times 2 are how 
many ? 

P. Four times 3 are how many ? Three times 4 are how 
many ? 

i#. If one postage stamp costs 2 cents, what will 6 cost ? 
How many 2's in 12 ? 

11. If one head of clover has 3 leaves, how many leaves 
will 4 heads of clover have ? How many 3's in 12 ? 

12. 2 flies have 12 legs. How many legs has 1 
fly ? How many 6's in 12 ? 12 legs divided be- 
tween 2 flies = ? 12 divided by 2 = ? One half 
of 12 = ? 

13. 3 bees have 12 wings. How many wings 




has each 



bee ? How many 4's in 12 ? 12 divided by 3 = ? One 
third of 12 = ? 

ljf. 4 ships have 12 masts. How many masts has 1 ship ? 
How many 3's in 12 ? 12 divided by 4 = ? One fourth 
of 12 = ? 

15. 6 flies have 12 wings. How many wings 
has 1 fly ? How many 2's in 12 ? 12 divided 
by6=? 

16. How many newspapers can be bought for 
12 cents, if they cost 2 cents each ? 3 cents ? 4 cents ? 

17. What will be the cost of one third of a load of hay at 
$12 a load ? One quarter of a load ? 




8 + 5 = 13 




1 1 1 1 1 1 I 


1 






J 





46 THIRTEEN. 

13. THIRTEEN. XIII. 

ADDITION. 

103. 1. Tommie spends 10 cents 10 + 3 = 13 

for peaches and 3 cents for an 
orange. How much does he spend ? 

2. How much are 10 and 3 ? 
3 and 10 ? 9 + 4 = 13 

3. If Tommie had paid 9 cents for 
peaches and 4 cents for an orange, 
how much would both have cost ? 

4. How many are 4 and 9 ? 

5. Mr. Sims pays $5 for a hat 
and $8 for a coat. What does he 
pay for both ? 

6. Harry in working examples, 
had 6 answers right and 7 wrong. 
How many problems did he work ? 

7. Add : 

1. 10 + 3. 3. 8 + 5. 5. 6 + 7. 7. 4+ 9. 

2. 9 +4. 4. 7 + 6. g. 5 + 8. s. 3 + 10. 

8. If one pole is 9 ft. long, and another 4 ft., how long are 
the two poles together ? 

9. Henry saves 6 cents one day and 7 cents the next. 
How many cents does he save in the two days ? 

10. George walked 7 miles on Monday, and 5 miles on 
Tuesday. How far did he walk in the two days ? 

11. A slate costs 7 cents, a lead pencil 3 cents, and a 
sponge 3 cents. How much do they all cost ? 

12. Fred paid 2 cents for a pen and pen-holder, 8 cents 
for a bottle of ink, and 3 cents for a rubber. How much did 
they cost him ? 

13. Jennie has 2 five-cent pieces, and a three-cent piece. 
How much money has she ? 



7 + 6 = 13 






! 1-1 1 1 II 

















THIRTEEN. 47 



104. State 
1. 


sums 

9. 


«< sigrM 


s. 




4. 


5. 


3 


4 




5 




6 


7 


10 


9 




8 




7 


6 


6. 


r. 




s. 




9. 


JO. 


4 


3 




7 




5 


8 


8 


8 




5 




6 





105. Write from dictation and add : 




l. 




2. 




3. 




4. 


35 




17 




9 




41 


24 




20. 




21 




23 


13 




42 




30 




15 


21 




13 




11 




14 


5. 




6. 




7. 




s. 


76 




59 




8 




27 


12 




24 




63 




43 


3 




10 




11 




12 



WRITTEN PROBLEMS. 

106. i. What is the sum of 24, 32, 14, and 23 ? 

2. On four shelves there are 16, 20, 42, and 15 books. How 
many books are there on the four shelves together ? 

3. If a steamboat sails 17 miles, 15 miles, 21 miles, and 
20 miles in four different hours, how far will it sail in the 
four hours together ? 

Jj,. Samuel earns $12 a week ; his brother, Jacob, $18 ; and 
his father, $43. How much do they together earn ? 

5. On four different farms there are 23, 14, 16, and 30 
cows. How many cows altogether are there ? 

6. A farmer had 38 quarts of blackberries, 14 quarts of 
raspberries, and 21 quarts of currants. How many quarts of 
the different kinds of berries together did he have ? 



48 



THIRTEEN. 



13. THIRTEEN. XIII. 



SUBTRACTION. 



13 - 3 = 


10 


13 - 


10 = 3 


13 - 4 = 


9 


13 - 


9 = 4 


13 - 5 = 


8 


13 - 


8 = 5 


13 - 6 = 


7 


13 - 


7 =e 



107. L If I earn $10 
and $3, how many dollars 
do I earn ? 

2. If I earn $13 and 
spend $3, how many dol- 
lars shall I have left ? 

3. From 13 take 3. 
How many remain ? 13 — 10=? 

Jf. 13 minus 5 equals how many ? 

5. 13 birds minus 8 birds = ? 

6. How many are 4 and 9 ? 4 from 
13 leave how many ? 9 from 13 ? 

7 . How many are 5 and 8 ? 5 
from 13 leave how many ? 8 from 
13? 

8. How many are 6 and 7 ? 6 
from 13 leave how many ? 7 from 13 ? 

9. If Fred rows 8 miles up a river and then 5 miles further, 
how far will he be from where he started ? 

10. If Fred then rows 5 miles back, how far will he be 
from where he first started ? 

11. If I buy 6 cents' worth of tape and 13 cents' worth of 
braid, how much more does the braid cost ? 

12. Carrie is 13 years old ; her little sister, Edith, is 4 
years old. How much older is Carrie ? 

13. Minnie had 12 cents in her bank. She spent 5 cents. 
How many cents did she then have ? 

ljj,. Mabel had 13 cents in her bank. She spent 5 cents. 
How many cents did she then have ? 

15. Susie is now 13 years old. She has attended school for 6 
years. How old was she when she began to attend school ? 






THIRTEEN. 






. State differences at sight : 






1. 2. 


3. 


4. 


B. 


13 13 


13 


13 


13 


10 9 


8 


7 


6 


6. 7. 


8. 


9. 


10. 


13 13 


13 


13 


13 


5 4 


3 


2 


1 



49 



$33 

6 



Arts. 



WRITTEN PROBLEMS. 

109. i. Mr. Clark earns $33 per week. He spends $17 
for board. How much has he left ? 

Explanation. — 1. Under $33 write 17, plac- 
ing units under units and tens under tens. 

2, 7 units cannot be taken from 3 units; 1 

ten is taken from the 3 tens ; it is changed to 

units, making 10 units, and then it is added to 

the 3 units, making 13 units ; 7 units from 13 

units leave 6 units. Write 6 in units' place. 

3, 1 ten having been taken from the 3 tens, 2 tens are left ; 1 ten 

from 2 tens leaves 1 ten. Write 1 in tens' place and write the dollar 

sign before it. Answer, $16. 

2. A history cost 72 cents ; a reader, 48 cents. The his- 
tory costs how much more than the reader ? 

3. A box of paper costs 33 cents ; a package of envelopes 
costs 15 cents. Which costs more, and how much ? 

110. Write from dictation and subtract : 



1. 


<> 


3. 


4. 


82 


53 


23 


43 


19 


24 


18 


27 


5. 


6. 


7. 


s. 


73 


61 


92 


33 


55 


49 


35 


18 



50 FOURTEEN. 

14. FOURTEEN. XIV. 

ADDITION. 

111. 1. Sailing a ship are 10 
men and 4 boys. How many men 
and boys are there together in the 
crew ? 4 and 10 are how many ? 

2. Sailing a second ship are 9 



10 + 4: 




14 




Mini 





















I 



II II 1 1 


1 1 






! i 





men and 5 boys. How many men 
and boys are there together in its 
crew ? 5 and 9 are how many ? s + C — 1 4- 

3. In a grocery, there are 8 bar- 
rels of flour and 6 barrels of sugar. 
How many barrels of both are 
there ? 7 + 7 = 14 

Jf. Of the horses in a stable, 7 
are white and 7 are black. How 
many horses, white and black, are 
there in the stable ? 

5. In a yard there were 2 roosters, 8 white hens, and 4 red 
ones. How many chickens together were there in the yard ? 

112. Copy and complete: 



i. 10 + 4 = 


3. 


8 + 6 = 




s. 


6 + 8 = 


7. 4 + 10 


g. 9 + 5 = 


4. 


7 + 7 = 




a. 


5 + 9 = 


*. 3 + 11 


113. Write from dictation and add: 




l. 


3. 




3. 




4. 


5. 


4 


1 




4 




2 


7 


2 


5 




2 




6 





3 


4 




6 




4 


3 


4 


4 




1 




2 


4 



ion and add: 




3. 4. 


B. 


10 41 


37 


36 17 


14 


43 20 


11 


5 5 


22 



51 



1. 2. 

24 16 

30 24 

15 10 

.22 23 

WRITTEN PROBLEMS 

115. 1. What will be the cost of some calico at 14 cents ; 
muslin, 23 cents ; linen, 47 cents ; and ribbon, 10 cents ? 

2. Amanda went to the grocery and paid 32 cents for cof- 
fee, 24 cents for sugar, 18 cents for cheese, and 20 cents for 
syrup. How much was her bill ? 

3. In a pasture were grazing 40 sheep, 12 horses, 15 
cows, and 7 oxen. How many animals were grazing in the 
pasture ? 

J/,. In a factory there worked 27 men, 12 women, 33 boys, 
and 10 girls. How many persons were working in the fac- 
tory ? 

5. Mr. Henry sold 25 spellers, 13 readers, 14 geographies 
and 21 histories. How many books did he sell ? 

6. A merchant sells in three days goods worth $20, $45, 
and $29. How much money should he receive for the goods 
sold in the three days together ? 

7. A lady paid $34 for a watch, $28 for a chain, $15 for a 
pair of bracelets, and $7 for a locket. How much did she 
pay for all together ? 

8. A gentleman bought a suit of clothes for $48, an over- 
coat for $33, and a hat for $3. How much money did he 
spend ? 

9. On a farm there are 25 acres of corn, 15 acres of wheat, 
and 13 acres of oats. How many acres altogether are there ? 

10. A farmer sells 40 bushels of apples, 22 bushels of 
potatoes, and 12 bushels of turnips. How many bushels 
altogether does he sell ? 



52 



FOURTEEN. 



14 - 4 = 10 


14 - 10 = 4 


14 - 5 = 9 


14 - 9 = 5 


14 - 6 = 8 


14 - 8 = 6 




14 - 7 = 7 





14. FOURTEEN. XIV. 

SUBTRACTION. 

116. i. In a piece of 
cloth are a dress pattern 
of 10 yards and 4 yards 
more. How many yards 
of cloth in the piece ? 
10 and 4 are how many ? 

2. If 4 yards are cut 
from the piece, how many yards will remain ? 4 from 14 
leave how many ? If 10 yards were cut off, how many would 
remain ? 10 from 14 leave how many ? 

3. In a ring are 14 marbles. Charlie knocks 5 of them 
out. How many are left ? 

J/,. Frank saw 14 birds on a telegraph wire. He shot 6 of 
them. How many escaped ? 

5. Fannie had 14 cents. She spent 8 cents for needles 
and 5 cents for candy. How many cents did she then have ? 

6. If of 14 cakes, a baker sells 9, how many cakes will 
he have left ? 

7. If a grocer having 13 pounds of tea sells 5 pounds and 
3 pounds, how many pounds will remain ? 

8. 14 is how much greater than 6 ? 



117. State differences at sight: 



1. 


2. 


3. 


4. 


5. 


14 


14 


14 


14 


14 


7 


4 


8 


5 


9 


6. 


7. 


8. 


9. 


10. 


14 


13 


12 


11 


14 


6 


7 


8 


9 


10 



FOURTEEN. 53 

118. Write from dictation and subtract : 



1. 


3. 


3. 


4. 


94 


63 


74 


82 


69 


28 


35 


57 


s. 


6. 


7. 


8. 


43 


84 


64 


34 


16 


37 


46 


28 



(ji 



66 



WRITTEN PROBLEMS. 

119. i. In a company of soldiers there were 94 men. 
After a battle, it was found that 28 men had been killed. 
How many men were there then in the company ? 

Explanation. — 1. Write 28 under 94, placing 
men units under units and tens under tens. 

& 2. Since 8 units cannot be taken from 4 

units, take 1 ten or ten units from the 9 tens, 
and add it to the 4 units, making 14 units ; 8 
men. units from 14 units leave 6 units. Write 6 in 
units' column. 
S. Since 1 ten was taken from the 9 tens, leaving 8 tens, take 2 
tens from 8 tens and 6 tens will remain. Write 6 in tens' column. 
Answer, 66 men. 

2. A bedstead cost $44 ; a lounge cost $15 less. How 
much did the lounge cost ? 

3. A horse was sold for $74, and this was $27 more than 
the sum for which the wagon was sold. For how much was 
the wagon sold ? 

Jf. Some lumber was bought for $38 and sold for $53. 
How much money was made by the sale ? 

5. Henry caught in one week 74 trout ; John caught in 
the same time 29 trout. How many more trout did Henry 
catch ? 

6. How much greater is 94 than 19 ? 



54 FOURTEEN. 

14. FOURTEEN. XIV. 
MULTIPLICATION AND DIVISION. 



2 x 7 = 14 

7 x 2 = 14 



14 -f- 2 = 7 
14 ^ 7 = 2 



120. 1. In each of 2 
schooners there are 7 sail- 
ors. How many sailors are 
there in both schooners ? 

2. On each of 7 trees there were 2 bushels of pears. How 
many bushels of pears were there on the 7 trees ? 

3. 7 + 7=? 2 + 2 + 2 + 2 + 2 + 2 + 2=? 
Jf. Two 7's are how many ? Seven 2's are how many ? 

5. 2 times 7 are how many ? 7 times 2 are how many ? 

6. 2x7= ? 7x2= ? 

7. Twice 7 are how many ? 

8. If 14 sailors are divided equally between 2 ships, how 
many sailors will there be to each ship ? How many 7's in 14 ? 

9. If 14 bushels of pears grow on 7 trees, how many bushels 
should grow on one of the trees ? How many 2's in 14 ? 

10. 14 divided by 2 are how many ? 14 divided by 7 are 
how many ? 

11. 14—2= ? 14+7= ? 

12. One half of 14 is how many ? 

13. How far will 2 chains extend, if each is 7 feet long ? 
IJj,. Ambrose can walk two miles an hour. How far can 

he walk in 7 hours ? 

15. A horse traveled 14 miles in 2 hours. How far did he 
travel in half the time ? 

16. A horse was given 14 quarts of oats in a week, or 7 
days. How much was this for each day ? 

17. What will be the cost of 7 postage stamps at 2 cents 
each ? 

18. A rod of iron is 14 ft. long ; it is 7 times as long as 
another rod. How long is the other rod ? 



FIFTEEN. 



55 



15. FIFTEEN. XV. 



ADDITION. 

121. 1. In going an errand, 
George paid 10 cents fare on the 
elevated road, and, in returning, 
5 cents fare on the surface line. 
How much did the two rides cost ? 
5 and 10 are how many ? 

2. Katie saw 9 sheep and 6 cows 
in one pasture. How many sheep 
and cows together did she see in the 
pasture ? 6 and 9 are how many ? 

3. Hattie owns 7 books and 
Carrie, 8 books. How many 
do they both own ? 

Jf. If Edward walks 6 miles 
and rides 9 miles, how far 
does he travel ? 

122. Announce sums at sight: 



10 + 5 = 15 



Mill 
























9 + 6 




: 15 


rm rr 






1 


8 + 7 










= 15 


Mill 
























hbbii 



Read answers : 
top to bottom ; 4. 

1st. 2d. 

( 8 4 

A ) 5 10 



1. From left to right ; 2. From right to left ; 3. From 
From bottom to top ; 5. As directed by the teacher. 

3d. 4th. 5th. 6th. 7th. 8th. 9th. 

5 10 8 9 8 6 10 

10 2 6 3 7 7 3 



















■ 


7 


9 


7 


9 


3 


8 


10 


8 


7 


7 


4 


6 


6 


10 


4 


4 


3 


5 


3 


10 


7 


5 


9 


4 


5 


7 


9 


9 


5 


7 


7 


5 


7 


8 


8 


2 


4 


6 


6 


6 


5 


2 


4 


5 


5 


8 


8 


6 


9 


6 


10 


9 


9 


5 



56 




FIFTEEN. 






123. 


Write from dictation and add : 






i. 


2. 


3. 


4. 


5. 


6. 


4 


5 


2 


5 


8 


6 


6 


4 


5 


1 


2 


1 


2 


1 


4 


2 


1 


3 


1 


5 


3 


4 


4 


5 



124. Write from dictation and add : 



1. 


2. 


3. 


4. 


B. 


35 


14 


23 


19 


36 


27 


25 


32 


31 


12 


11 


36 


15 


30 


14 


20 


10 


12 


4 


13 



Note. — In adding the above think and announce the results only ; 
thus, in Ex. 1 : Units, 8, 13 ; tens, 3, 4, 6, 9. 

WRITTEN PROBLEMS. 

125. i. Three blocks of houses contain 24, 15, and 26 
houses. How many houses are there in the three blocks ? 

2. Four farms contain 37, 21, 16, and 20 acres. How 
many acres are there in the four farms together ? 

3. In four days Mr. Myers walked 25, 13, 32, and 25 miles. 
How far did he walk altogether ? 

Jf. Four books cost 25 cents, 22 cents, 18 cents, and 20 
cents. How much did the four books together cost ? 

5. In a park there are 18 boys playing ball, 12 boys flying 
kites, and 24 boys playing marbles. How many boys alto- 
gether are there ? 

6. A farmer bought a cow for $28, a calf for $12, and a 
sheep for $14. How much did he pay for all ? 

7. What will be the entire cost of 4 pairs of shoes at $2 
a pair ; 5 pairs of slippers at $1 a pair, and a pair of boots 
worth $12 ? 



FIFTEEN. 



5? 



15. FIFTEEN. XV. 



SUBTRACTION. 



126. i. Jacob learned 
to swim when 15 years 
old. He learned to skate 
when he was 5 years 
younger. How old was 
he when he learned to 
skate ? 

2. 5 from 15 leave how many ? 



15 - 5 = 


10 


15 - 


10 = 5 


15 - 6 = 


9 


15 - 


9 = 6 


15 - 7 = 


8 


15 - 


8 = 7 



10 from 15 ? 

9 from 15 ? 

8 from 15 ? 



3. 6 from 15 leave how many ? 
4- 7 from 15 leave how many ? 

5. Margie lived in the city of Buffalo for 7 years. She 
then, at the age of 15, moved with her parents to St. Louis. 
How old was she when she first lived in Buffalo ? 

6. Ella has 15 yards of ribbon. She cuts off 8 yards and 
gives them to her sister. How many yards has she left ? 

7. Bertie bought 9 tops ; he lost 3 tops ; and then his 
brother gave him 7. How many tops has he now ? 

8. Tom has 8 marbles and Ned, 7 marbles. They lose 
6 marbles. How many have they then ? 



127. Jfliat is the value of: 



1. 15 — 5 ? 

2. 15 — 6? 

3. 15 — 7? 



4. 15— 8? 

5. 15— 9? 

6. 15 — 10? 



7 . 3 x 3 + 3_5? 

8. 4x3 + 2-6? 
9 12-f-4 + 2-4? 



128. State differences at sight. 



1. 


2. 


3. 


4. 


5. 


6. 


15 


15 


15 


15 


15 


15 


5 


6 


7 


8 


9 


10 



58 




FIFTEEN. 






129. Write front dictation and subtract : 




1. 


3. 




3. 


4. 


85 


74 




65 


43 


16 


26 




37 


28 


5 


6. 




7. 


8. 


92 


55 




45 


74 


54 


39 




18 


29 



WRITTEN PROBLEMS. 

130. i. A baker had 65 rolls. He sold 36. How many 
remained unsold ? 

2. A horse trotted 45 miles one day and only 17 miles the 
next day. How much farther did he travel on the first day ? 

3. In a class are 44 pupils ; 27 of them are girls. How 
many of the pupils are boys ? 

4- To cover a floor with carpet costs $73, and to cover it 
with matting costs $16. How much less does the matting 
cost ? 

5. Mr. Hartley buys shades for $27 and rugs for $18. He 
pays $29 cash. How much still remains unpaid ? 

6. 85 men worked in a foundry. After 17 were discharged, 
how many men worked there ? 

7. A man bought a horse for $75 and sold it for $56. How 
much money did he lose by the sale ? 

8. A jeweler bought a watch for $38, and a chain for $15. 
He sold watch and chain together for $65. How much 
money did he gain ? 

9. Ella has 75 cents ; she spends 38 cents for ribbons. 
How much money has she left ? 

10. How much more will 7 hats at $2 each cost than 4 
hats at $3 each ? 

11. Henry has a book containing 85 pages. He has read 
27 pages. How many pages has he yet to read ? 

12. How much greater is 75 than 57 ? 



FIFTEEN. 



59 



15. FIFTEEN. XV. 



MULTIPLICATION AND DIVISION. 



3 x 5 = 15 

5x3=15 



15 -f- 3 = 5 
15 -*- 5 = 3 



131. 1. At 5 cents each, 
what will be the fare for 3 per- 
sons riding in a street car ? 

2. At $3 each, what will be 
the charge for 5 passengers sailing over Long Island Sound ? 

5.5 + 5 + 5=? 3 + 3 + 3 + 3 + 3 = ? 

J^. Three 5's are how many ? Five 3's are how many ? 

5. 3 times 5 are how many ? 5 times 3 are how many ? 

6. 3x5 = ? 5x3=? 

7. At 5 cents each, for how many rides in a street car will 
15 cents pay ? How many 5's in 15 ? 

8. At $3 each, for how many trips on a steamboat will 
$15 pay ? How many 3's in 15 ? 

9. 15 divided by 3 are how many ? 15 divided by 5 are 
how many ? 

itf.3x5=? 15—3= ? 5x3=? 15-^5=? 





11. One third of 15 is how many ? One fifth of 15 ? 

12. The maple leaf has 5 lobes or parts. How many lobes 
have 3 leaves ? 

13. 3 pansies have 15 petals, or leaves. How many petals 
has each pansy ? 

1]±. Carrie bought 5 bananas at 3 cents each. How much 
did they cost ? 



60 



SIXTEEN. 



16. SIXTEEN. XVI. 



ADDITION. 



132. 1. Emily hemmed 10 hand- 10 + 6 = 16 

kerchiefs one day and 6 the next. 
How many handkerchiefs did she 
hem in the two days ? 6 and 10 
are how many ? 

2. Sadie had 9 oranges and May 
had 7. How many oranges did 
both have ? 

3. In a park were 8 large elms 
and 8 small ones. How many elms 
were there in the park ? Two 8's 
are how many ? 

^. Harry counted the birds on four different trees and 
found that there were 5, 4, 3, and 4. How many birds were 
there altogether ? 

5. 7 cents and 9 cents together are how much greater than 
10 cents ? 



MM 










, 
















<i 


9 + 


7 = 16 


MM 
























8 + 


8 = 16 


MM 





























133. Copy and complete: 

i. 10 + 6= 4. 7+ 9= 7. 4 + 8= to. 5 + 8: 

2. 9 + 7= 5.6 + 10= *. 3 + 7= n.6 + 9: 

3. 8 + 8= 6. 5 + 9 = 9. 2 + 9= 12. 8 + 7 



13. 3 x5 — 5 + 1 = 

14. 2x4 + 8 — 9 = 



is. 7 + 7 = 
is. 6 + 6 = 



17. 
18. 



15^5 + 4 + 9 

7x2-5 + 7 



134. State sums at sight: 



l. 

10 
6 



2. 

9 

7 



3. 

8 
8 



4. 

7 
9 



5. 

6 
10 



SIXTEEN. 61 

135t Write from dictation and add: 



1. 


a. 


3. 


4,. 


40 


39 


58 


17 


26 


27 


38 


69 


5. 


e. 


7. 


s. 


17 


23 


17 


10 


20 


35 


20 


35 


9 


13 


37 


18 


40 


24 


12 


23 



WRITTEN PROBLEMS. 

136. 1. In January there are 31 days ; in February, 28 
days ; and in March, 31 days. How many days are there 
in the three months ? 

2. Mr. Samuels paid $37 for an overcoat, $43 for a suit of 
clothes, and $16 for a pair of fine boots. How much did he 
pay for all ? 

3. Miss Lee spent for ribbons 36 cents, for laces 28 cents, 
for worsted 12 cents, and for pins 10 cents. How much 
money did she spend ? 

Jf. Tom had of different kinds of postage stamps 25, 31, 
16, and 24. How many did he have altogether ? 

5. A paper carrier delivered of different newspapers 14, 
22, 31, and 29. How many papers did he deliver ? 

6. How many fruit trees are there in an orchard contain- 
ing 17, 28, 21, and 30 of different kinds ? 

7. A car company has in four different stables, 28 horses, 
15 horses, 23 horses, and 20 horses. How many horses al- 
together has it in the four stables ? 

8. How many passengers are there in four cars, if the dif- 
ferent cars contain 20, 22, 24, and 26 passengers ? 

9. Find the sum of 11, 22, 33, and 20, 

10. Add 5 x 2 to 4 x 4, 



62 



SIXTEEN. 



16. SIXTEEN. XVI. 



16 - 


6-10 


16-10-6 


16- 


7=9 


16 - 9-7 






16-8-8 





SUBTRACTION. 

137. 1. Henry had 10 
cents ; his father gave him 
6 cents. How many cents 
did he then have ? 

2. If Henry spends the 
6 cents for tops, how many cents will he have left ? If he 
spends the 10 cents, how many will remain ? 

3. 6 from 16 leave how many ? 10 from 16 leave how many ? 
Jf, 7 from 16 leave how many ? 9 from 16 leave how many ? 

5. 8 from 16 leave how many ? How many 8's in 16 ? 

6. Two pans contain 9 quarts and 7 quarts of milk. After 
8 quarts are poured out, how many quarts are left ? 

7. From a mixture of grain containing 9 bushels of wheat 
and 6 bushels of corn, 7 bushels are taken. How many 
bushels of grain will be left in the mixture ? 

8. How much of $16 will remain after paying a bill of $7 ? 

9. Edith has a dime, or ten-cent piece, a nickel, or five- 
cent piece, and a penny. She spends 8 cents. How many 
cents has she left ? 

10. Mr. Moore has $16. He spends $8 for shoes and $2 
for a hat. How many dollars has he left ? 

11. Mr. Brown caught 16-fish. His two sons caught 5 fish 
and 3 fish. He caught how many more than the two sons ? 



138. Copy and complete : 



1. 


1G — 6= 4. 16— 9= 7. 16 — 9 


2. 


16 — 7= s. 16 — 10= s. 16 — 5 


3. 


16_8= 6. 16- 7= o. 16-8 




io. 5x3 + 1 — 8= j* 15-^-5 + 13 — 3 = 




ii. 2x7 + 2—9= is. 14+2+ 9 — 6 = 







SIXTEEN. 






!• State differences at sight: 






1. 


2. 


3. 


4. 


B. 


16 


16 


16 


16 


16 


10 


9 


8 


7 


6 


6. 


7. 


8. 


9. 


10. 


55 


46 


95 


45 


87 


25 


36 


50 


15 


67 



63 



WRITTEN PROBLEMS. 

140. 1. In a pond there were 76 fish ; 38 were caught. 
How many remained in the pond ? 

2. From a class of 46 pupils, 27 were promoted. How 
many w'ere left in the class ? 

3. Two rows of seats contained 17 and 19 pupils ; 6 pupils 
in one row and 12 in the other stood up. How many pupils 
remained seated ? 

I±. Mr. Perkins started on his vacation trip with $86. He 
spent $27 for car fare. How many dollars did he then have ? 

5. How much greater is 75 than 29 ? 

6. There were 36 sheep in a flock ; 19 of the sheep were 
sold. How many remained in the flock ? 

7. A man earned $86 a month. His expenses were $48. 
How much money did he save in a month ? 

8. What is the difference in the cost of 5 bananas at 3 cents 
each, and 4 oranges at 2 cents each ? 

9. Frank has a fifty-cent piece, a quarter or twenty-five 
cent piece, and a penny. If he spends 56 cents, how much 
money will he have left ? 

10. A railroad train travels 36 miles one hour and only 
29 miles the next hour. How much farther does it travel the 
first hour ? 

11. A man's horse cost him $96 ; his wagon cost him $69. 
How much more did the horse cost ? 



64 



SIXTEEN. 



2 x 8 = 16 
8 x 2 = 16 


16 -r- 2 = 8 

16 4- 8 = 2 


4 x 4 = 16 


16 -*- 4 = 4 




16. SIXTEEN. XVI 

MULTIPLICATION AND DIVISION. 

141. 1. In each of 2 boats 
there are 8 men. How many 
men are there in the two 
boats ? Twice 8 are how 
many ? 

2. Add: 

i. 8 + 8. 2. 4 + 4 + 4 + 4. 

8. 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2. 

3. Two 8's = ? Eight 2's = ? 
Jj,. Four times 4 oranges = ? 

Four 4's = ? 4x4=? 

5. 2x8=? 8x2=? 

6. At 4 cents each how much 
are 4 oranges worth ? 

7. There are 2 men in each of 8 boats. How many men 
are there in the 8 boats ? 

8. If there were 4 men in each of 4 boats, how many men 
would there be in the four boats ? 

9. If 16 men are divided into 2 equal boat crews, how 
many men will there be in each crew ? How many 8's are 
there in 16 ? 

10. If 16 men are divided into 4 equal boat crews, how 
many men will there be in each crew ? How many 4's in 16 ? 

11. If 16 men are divided into 8 equal boat crews, how 
many men will there be in each crew ? How many 2's in 16 ? 

12. 16 divided by 2 = ? 16 divided by 4 = ? 16 divided 
by 8= ? 

IS. 16-^-2 = ? 16-f-4 = ? 16-f-8 = ? 
ljf. One half of 16 is how many ? One fourth ? 
15. In 2 gallons of milk there are 16 pints. How many 
pints are there in one gallon ? 



SIXTEEN. 65 

142. Copy and multiply: 



S2 $ Explanation.— 1. Three 2's make 6. Write 6 

in units' place. 

2. Three 3's make 9. Write 9 in tens' place. 
Answer, 9 tens and 6 units, 9 dimes and 6 cents, 
or 96 cents. 



3 



q6* 



2. 3. 

$23 $22 $41 $13 21^ 

2 4 2 3 4_ 



7. 8. 9. lO. 11. 

120 110 100 44 34 

4 5 5 2 2 



12. 13. 14. IS. 16. 

33 20 31 33 24 
4 4 3 2 2 



4. 


5. 


$41 


$13 


2 


3 


9. 


10. 


100 


44 


5 


2 


14. 


is. 


31 


33 


3 


2 



143. Copy and divide: 



3)(j6t 



1- Explanation. — 1. One third of 9 tens is 

3 tens. Write 3 under the tens. 

2. One third of 6 units is 2 units. Write 
2 under the units. Answer, 3 tens and 2 
units, 3 dimes and 2 cents, or 32 cents. 



32 



2. 3. 4. S. 6. 

5)50 4)88 4 )80 3)99 3)63 

T. 8. 9. 10. 11. 

4)44 3 )69 2 )86 3}36 2)62 

12. 13. 14. 15. 16. 

2)46 3)93 2)66 4)84 2)82 



66 



SEVENTEEN. 



WRITTEN PROBLEMS. 

144. 1. If a Second Reader costs 32 cents, what will 3 cost ? 
2. ' If a Fourth Reader costs 43 cents, what will 2 cost ? 
<5. If a quire of paper costs 22 cents, what will 4 quires 
cost ? 3 quires ? 
Jf. If an overcoat costs $11, what will 5 overcoats cost ? 

5. At 14 cents a pound, what will 2 pounds of candles cost ? 

6. If a book-slate is worth 40 cents, what will 2 be worth ? 

7. If 3 Second Readers cost 96 cents, what will one cost ? 

8. If a man earns $39 in 3 days, how much does he earn 
in one day ? 

9. If $55 are spent for 5 weeks' boarding, how much is 
spent for one week ? 

10. If 2 dozen eggs cost 48 cents, what will one dozen cost ? 

11. 48 eggs are 4 dozen eggs ; how many eggs in 1 dozen ? 

12. Mr. Bryon is 84 years old. His son is only half as 
old. How old is his son ? 



17. SEVENTEEN. XVII. 



ADDITION. 



145. 1. Kittie is 10 years old. 10 + 7 = 17 

Her brother, William, is 7 years 
older. How old is William ? 

2. If Kittie were 9 years old and 
William were 8 years older, how 
old would he be ? 

3. In one mine there are 9 men working ; in another, 8 
men. How many men are working in both mines ? 

4- A young man earns 7 dollars a week and his father 
earns 10 dollars. How much do both earn ? 



r 




1 






• 


9 














+ 8 = 17 




























_\ 



SEVENTEEN. 67 

146. What is the value of: 

1. 6 + 8 + 2-3 ? 3. 10 + 7 ? 5. 13—3 + 7—5 ? 

2.5 + 7 + 3-4? 4. 9 + 8? 6. 9 + 7 — 5 + 4? 

147. Write from dictation and add : 



1. 


». 


3. 


4. 


s. 


5 


4 


9 


8 


4 


8 


5 





5 


3 


3 


4 


3 





5 


1 


3 


4 


4 


5 


6. 


7. 


«. 


9. 


10. 


27 


16 


50 


16 


20 


45 


37 


19 


38 


47 


14 


40 


28 


23 


30 



WRITTEN PROBLEMS. 

148. i. On one tree there are 37 apples, on another 24, 
and on another 16. How many apples are there on the three 
trees ? 

2. A lady paid 45 cents for berries, 24 cents for asparagus, 
and 17 cents for corn. How much did she spend ? 

3. A little girl bought a book for 28 cents, a pencil case 
for 27 cents, and a slate for 22 cents. How much did she 
pay for all ? 

J/,. In a factory there were 17 tons of nut coal, 20 tons of 
stove coal, and 30 tons of egg coal. How many tons of coal 
altogether w r ere there in the factory ? 

5. A girl picked 15 quarts of strawberries, 25 quarts of 
raspberries, and 47 quarts of blackberries. How many quarts 
of berries did she pick ? 

6* What is the sum of 3 x 3 and 4x2? 



68 



SEVENTEEN. 



17. SEVENTEEN. XVIL 

SUBTRACTION. 



17-7 = 10 1 17-10 = 7 



17 



9 



17- 9 = 8 



Take 9 ; how 



149. i. From 17 take 7; 
how many remain ? Take 
10. 

2. From 17 take 8; how many remain? 
many remain ? 

3. If a house is 17 feet wide and its side yard is 8 feet 
narrower, how wide is the side yard ? - 

Jj,. A fence is 16 feet high. If it were 8 feet lower, what 
would be its height ? 

5. Edward has a dime, or ten-cent piece, a five-cent piece, 
and a two-cent piece. He spends 11 cents. How many 
cents has he left ? 

6. If a man has $17 and pays $4 for each of 4 hats, how 
many dollars will he then have ? 



150. What is the value of: 



i. 17- 7? 

2. 17-10? 

3. 17- 8? 

4. 17— 9 ? 



5 . i4 + 3_2? 

e. 13 + 4-2? 

7 . ii + 6 — 8? 

8. 12 + 5-9? 



9. 2x8+ 1-10? 

10 . 16-^4 + 13-7? 

n. 8x2+ 1— 9? 

12. 16-^8 + 15— 8? 



151 . Write from dictation and subtract : 



1. 


9. 


3. 


4. 


s. 


37 


57 


97 


77 


96 


19 


38 


60 


59 


27 


6. 


7. 


s. 


9. 


10. 


85 


64 


55 


87 


67 


36 


37 


48 


39 


18 



EIGHTEEN. 



69 



18. EIGHTEEN. XVIII. 



ADDITION. 



1 1 


































9 + 9 = 18 


1 1 

































152. 1. A man paid $10 for 10 + 8 = 18 

one book and $8 for another. 
How much did he spend ? 8 and 
10 are how many ? 

2. 9 and 9 are how many ? Two 
9's make how many ? 

3. How far does a man run in 
two hours, if he runs 9 miles the first hour, and 9 miles the 
second hour ? 

Jf. Two 9's are how many ? Twice 9 are how many ? 

5. How far does a man walk in three hours, when in the 
first hour he walks 7 miles, in the second hour 3 miles, and 
in the third hour 8 miles ? 



153. What is the sum of: 



lt 9 + o + 9? 

2. 5 + 5 + 8? 

3. 6 + 7 + 5? 



4. 10+ 8? 

5. 9+9? 
e. 8 + 10? 



s. 2 + 7 + 9? 
9. 9 + 5 + 4? 



154. Write from dictation and add : 



1. 


s. 


3. 


4. 


6. 


39 


48 


27 


49 


16 


26 


21 


32 


24 


24 


13 


19 


29 


15 


38 



Note. — In adding the different columns, think of two smaller num- 
bers as being the number, to which, when combined, they are equal. 
Thus, in Ex. 1 : In units' column think simply, 9, 18 ; in tens' column, 
2, 7. Answer, 78. 



70 EIGHTEEN. 

18. EIGHTEEN. XVIII 

SUBTRACTION. 
155. 1. In Mabel's bank 



18-8 = 10 



18-10 = 8 



18 -9 = 9 



were 1 dime and 8 pennies. 
How much did she have in 
her bank ? 

2. If Mabel spends 10 cents from the money in her bank, 
how much will she have left ? If she spends 8 cents ? 

3. 8 from 18 leave how many ? 10 from 18 ? 

4* There are 18 bananas in a bunch ; if 9 are eaten, how 
many will remain ? 

5. 9 from 18 leave how many ? How many 9's in 18 ? 

6. Carrie had 18 cents in her purse. She first spent 4 
cents and then 5 cents. How many cents did she have left ? 

7. Emma is 9 years younger than her sister Kate. If Kate 
is 17 years old, how old is Emma ? 

8. Howard in a hop, skip, and jump went 3 feet, 6 feet, 
and 9 feet. This was 9 feet farther than Edward could go 
in a hop, skip, and jump. How far could Edward go ? 

9. Sarah has a $10 bill, a $5 bill, and a $2 bill. If she 
spends $6 for shoes and $3 for gloves, how much money will 
she then have ? 

156. What is the value of: 

lm 16 + 2— 9? 4. 18- 8? 7. 7 + 11 — 9? 

2 , 15 + 3— 8? 5. 18— 9? 8. 6+ 5 — 8? 

3 . 14 + 4—10? 6. 18 — 10? 9. 8- 3 + 9? 

157. Write from dictation and subtract : 

l. is. 3. 4. 5. 

48 37 68 93 78 

29 18 19 56 18 



EIGHTEEN, 71 



WRITTEN PROBLEMS. 

158. 1. If Mr. Phillips is 58 years old and his son, 29 
years old, how much older is Mr. Phillips than his son ? 

2. After traveling 38 of the 73 miles from New York to 
New Haven, how many more miles must a man travel to 
complete the journey ? 

3. How much more is paid for a dress than for a bonnet, 
if the bonnet costs $19 and the dress $78 ? 

4- A grocer buys 88 bushels of potatoes and sells 29 bushels. 
How many bushels has he left ? 

5. A baker received 13 and 15 barrels of flour. How 
many did he have left, after using 9 barrels ? 

6. George had 37 marbles and lost 18. How many did he 
have left? 

7. A painting was bought for $88 and sold for $49. How 
much was lost ? 

8. If from 4 x 22 you subtract 3 x 23, how many will 
remain ? 

9. A man had $40, and $5, and $2. He spent $18. How 
many dollars did he then have ? 

10. Edward had saved $78. He spent $21, $13, and $15. 
How many dollars did he then have ? 

11. Each of 4 boys had $12. After buying a boat for $35, 
how many dollars did they together have ? 

12. Richard earned $47 in a month. He took in pay- 
ment a cow, worth $30, and the rest in money. How much 
money did he receive ? 

13. Mr. James earned in one week $36 and his son earned 
$22. They paid for rent $14, and for clothes $15. How 
much money did they together have left ? 

14^ How much greater is 4 x 22 than 3 x 11 ? 

15. From a piece of cloth containing 68 yards, there were 
sold 17 yards, 18 yards, and 15 yards. How many yards 
remained in the piece ? 



72 



EIGHTEEN. 



18. EIGHTEEN. XVIII. 

MULTIPLICATION AND DIVISION. 



2 x 9 = 18 
9 x 2 = 18 



3x6=18 
6 x 3 = 18 



18 -f- 2 = 9 
18 -i-9 = 2 



18 -f- 3 = 6 

18 -f- 6 = 3 



159. 1. In a game of 
base-ball there are 9 men 
on a side. How many men 
are there on the 2 sides ? 

2. Add: 

1. 9 + 9. 

2. 6 + 6 + 6. 

s. 3 + 3 + 3 + 3 + 3 + 3. 

4. 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2. 

3. Two 9's are how many ? Nine 2's are how many ? 
Jf. Six 3's are how many ? Three 6's are how many ? 

5. 2 times 9 are how many ? 9 times 2 are how many ? 

6. 2x9= ? ' 9x2= ? 

7. 6x3= ? 3x6=? 

8. How many gloves are there in 9 pairs of gloves, if two 
gloves make one pair ? 

9. A grasshopper has 6 legs. How many legs have 3 grass- 
hoppers ? 

10. 3 feet are equal to a yard. How many feet are there 
in 6 yards ? 

11. If 18 boys are divided into 2 equal sides in a game of 
base-ball, how many boys will there be on a side ? How many 
9's in 18 ? 

12. If 18 skates are divided among 9 skaters, how many 
skates will each skater have ? How many 2's in 18 ? 

13. If 18 cents buy 3 yards of elastic, what is the cost 
of one yard ? How many 6*s in 18 ? 

ljf> Mr. Frost earns $18 in a week of 6 working days. 
How much does he earn in one day ? How many 3's in 18 ? 

15. 18 divided by 2 are how many ? 18 divided by 9 are 
how many ? 



EIGHTEEN. 73 

16. 18 divided by 3 are how many ? 18 divided by 6 are 
how many ? 

17. What is one half of 18 ? One third ? 

18. 3 lilies have 18 petals. How many 
petals has one lily ? 

19. If 6 books cost $18, what will one book 
cost ? 

20. If Theodore can run 6 miles in an hour, 
how far at the same rate can he run in 3 hours ? 

21. In how many days can one man do as much 
work as can be done by 2 men in 9 days ? 

22. If half a yard of silk plush costs $9, what 
will a yard cost ? 

160. State products at sight : 




1. 


2. 


3. 


4. 


s. 


«. 


2 


4 


2 


3 


2 


3 


2 


2 


3 


3 


4 


2 



161. State quotients at sight: 



1. 


». 


3. 


4. 




5. 


2)8 


3)6 


4)4 


5)5 


2 


)J 


«. 


7. 


s. 


9. 




m 


2)4 


4)8 


3)9 


2)6 


3 


H 


162. W 


rite from 


dictation and 


niidtiply : 






l. 


2. 


3. 


4. 


5. 




11 


10 


11 


10 


11 




6 


8 


7 


7 


8 




6. 


7. 


«. 


9. 


10. 




32 


24 


31 


23 


44 




3 


2 


2 


3 


2 





74 EIGHTEEN. 

163. Write from dictation and divide : 

1. 2. 3. 4. 

2)42 3)69 4)48 5)55 



5. 6\ 7. 8. 

4 )80 2 )28 3 )93 2 ). 62 

WRITTEN PROBLEMS. 

164. 1. In an orchard there are 77 trees in 7 equal rows. 
How many trees are there in one row ? 

2. If 4 car drivers together receive $48 dollars per week, 
how much does each car driver receive ? 

3. What will be the cost of one yard of ribbon, if 3 yards 
cost 69 cents ? 

4- If a silver watch costs $21, at the same price what will 
be the cost of 3 silver watches ? 

5. When a canal boat is traveling at the rate of 32 miles a 
day, how far will it travel in 2 days ? 

6. If Mr. Jones earns $42 in one week, how much can he 
earn in 2 weeks ? 

7 . What will 4 pairs of blankets cost at $21 a pair ? 

8. If 2 pineapples cost 40 cents, what is the cost of one 
pineapple ? 

9. Mr. Briggs pays $33 for 3 months'' rent. How much 
does he pay for one month/s rent ? 

10. If Mr. Briggs paid $22 per month for rent, how much 
would he pay for 4 months' rent ? 

11. How many hats at $3 each can be bought for $99 ? 
For $69 ? For $36 ? 

12. Find the entire cost of 4 tables at $12 each and 3 
desks at $11 each. 

13. What will be the united cost of 5 book-cases at $10 
each and 3 rocking-chairs at $13 each ? 

14. 4 x 22 less 3 x 23 = ? 



NINETEEN. 



75 







10 + 9 = 




19 


) 




1 






1 































19. NINETEEN. XIX. 

ADDITION. 

1. In one farm there are 10 
acres, and in another, 9 ; how 
many acres are there in the two 
farms ? 

2. If there are 9 men in one boat and 10 men in another, 
how many men are there in both boats ? 9 + 10 = ? 

165. Announce sums at sight: 

Read answers : 1. From left to right ; 2. From right to left ; 3. From 
top to bottom ; 4. From bottom to top ; 5. As directed by the teacher. 



C- 



D 



1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 

986579974 
784491834 



3 


2 


9 


7 


6 


4 


3 


7 


2 


7 


5 


9 


4 


3 


5 


6 


2 


8 


4 


3 


7 


8 


6 


3 


2 


2 


4 


6 


8 


1 


9 


5 


5 


9 


7 


3 


3 


5 


8 


9 


3 


5 


6 


2 


3 


4 


3 


2 


6 


3 


2 


2 


3 


2 



166. Write from dictation and add : 



1. 


2. 


3. 


4. 


5. 


6. 


7 


4 


4 


3 


6 


1 


3 


6 


3 


■ 7 


3 


4 


3 


2 


2 





2 


2 


2 


4 


1 


5 


7 


8 



Note. — In adding columns, consider frequent combinations as single 
figures. Thus, in Ex. 1 : 5, 15 ; Ex. 2 : 6, 16 ; Ex. 5 : 9, 18. 



76 





NINETEEN. 




Write from dictation and add : 




1. 


2. 


3. 


4. 


25 


29 


14 


30 


24 


10 


32 


28 


31 


16 


24 


23 


15 


24 


26 


15 


s. 


6. 


7. 


8. 


21 


8 


6 


15 


35 


17 


27 


9 


14 


21 


13 


20 


10 


33 


20 


32 


3 


10 


12 


23 



WRITTEN PROBLEMS. 

168. 1. A horse cost $65 ; the harness, $15 ; a whip, $3 ; 
and a blanket, $10. What was the cost of all ? 

2. In four classes of pupils there are 14, 29, 20, and 16 
pupils. How many pupils are there altogether ? 

3. A bedstead cost $35 ; a rocking-chair, $16 ; a table, 
$12 ; a mattress, $11 ; and a pillow, $4. What was the 
amount of the bill ? 

4. If Mr. Jacobs earns $19, $36, and $24, and spends $50 
and $26, how much will he have left ? 

<5. Mr. Eoss owes $69. He pays $16, $22, and $18. How 
much does he still owe ? 

6. How many tons of coal did Mr. Eagleton sell to four 
families taking respectively 38, 25, 14, and 12 tons ? 

7. A clock cost $9 ; a lamp, $12 ; a pair of bronze vases, 
$23 ; and a match safe, $3. What did all cost ? 

8. A builder pays 3 men, $23 each ; and 2 men, $10 each. 
How much does he pay to the 5 men ? 

9. Find the sum of 30, 29, 10, and 20. 

10. What is the sum of 3 x 6 and 3 x 4 ? 



NINETEEN. 77 

19. NINETEEN. XIX. 

SUBTRACTION. 



19-9= 10 



19-10=9 



169. 1. A steel rod is 
19 inches long. If 9 inches 
are cut off, how long will the rod be ? 

2. Frank started to walk 19 miles. He rested after walking 
10 miles. How far had he yet to travel ? 

3. 19 — 10= ? 19 — 9= ? 

170. Announce differences at sight: 

1st. 2d. 3d. Jfth. 5th. 6th. 7th. 8th. 9th. 

(13 17 15 12 11 14 15 18 19 

A j 8 9 6 4 8 7 9 8 10 



(12 14 16 11 12 14 13 10 17 

B j 9 6 9 3 6 5 57 8 



(10 13 16 19 11 16 12 13 14 

C j 6 4 8 9 5 7 89 8 



\ 14 16 18 11 12 10 15 10 15 
D |_9_6_9 7 7 5 83 7 



171. Write from dictation and subtract: 



1. 


2. 


3. 


4. 


5. 


98 


37 


54 


62 


86 


49 


29 


38 


57 


48 


6. 


7. 


8. 


9. 


10. 


75 


49 


77 


94 


87 


58 


19 


59 


67 


59 



78 TWENTY. 

20. TWENTY. XX. 

ADDITION. 

172. 1. If a dress costs $56, a bonnet $16, and a parasol 
$8, what will the three together cost ? 



to 



/ Explanation. — 1. Write the numbers so 

^ fV»of nm'tc otonrl \ 



46 



that units stand in one column and tens in an- 
other. 

2. Units, 14, 20 ; or 2 tens and units. 
O Write in the units' column, and carry the 
2 tens to the tens' column. 

$80 AnS. 3 - Tens > 3 ' 8 * Write 8 in the tens ' column. 

8 tens, units, or 80. 

The dollar sign, $, written before the 8 gives the answer, $80. 

2. What will be the cost altogether of a suit of clothes, 
$44 ; an overcoat, $29 ; and a pair of boots, $13 ? 

3. Andrew has $20 in one cash box ; $15 in another box ; 
and $35 dollars in his purse. How much money altogether 
has he ? 

4* A man spent $17 for coal, $18 for a stove, and $25 for 
a bedstead. How much money did he spend ? 

173. Write from dictation and add : 



1. 


3. 


3. 


4. 


5. 


;24 


14^ 


37^ 


$18 


$25 


37 


28 


17 


37 


19 


29 


38 


26 


25 


36 


6. 


7. 


8. 


9. 


lo. 


12 


23 


12 


28 


10 


25 


16 


27 


14 


15 


23 


24 


13 


23 


25 


14 


15 


15 


12 


16 


16 


11 


23 


21 


24 



TWENTY. 79 

WRITTEN PROBLEMS. 

174. 1. Find the sum of $12, $18, $27, $13, $10. 

2. A pound of sugar costs 7 cents, a pound of coifee 
28 cents, a pound of lamb 

chops 20 cents, a pound of ji ^-^LILI^^ ^ ffjL 

porterhouse steak 25 cents, IfUtffr VAiW^i t^i %X \ 
and a pound of lard 10 cents. fljl -^ ^^^ j!f 

If a pound of each is bought, ^tJKsa^^^^^^^^» SJu» 
what will be the cost of all ? '** " 

<?. Find the cost altogether of a yard of ribbon, 20 cents ; 
a yard of muslin, 8 cents ; a yard of gingham, 12 cents ; 
a yard of cambric, 5 cents, and a yard of lace, 25 cents. 

4. If Mr. Knight has $55 and spends $38 of it for furni- 
ture, how much money will he then have ? 

5. Mr. Farmer bought a cow for $23, a calf for $8, and a 
horse for $48. He gave in payment eight $10 bills. How 
much change was coming to him ? 

6. A merchant had 35 gallons of oil in one tank, and 45 
gallons in another. After selling 39 gallons, how much oil 
did he have left ? 

7. A man paid $65 for a horse, and $25 for a cart. He 
sold the two together for $18 less than their cost. For how 
much did he sell them ? 

8. A sewing-machine cost $75 ; a cover for the machine 
cost $5. They were sold for $57. How much less than their 
cost was the selling price ? 

9. In one field there were 49 acres of land, in another field 
there were 30 acres. From the two fields 58 acres were sold. 
How many acres were left ? 

10. A man earns $20 a week for 4 weeks. He then pays 
out $25 for rent. How much money has he left ? 

11. Out of $90 a man pays $41 for a suit of clothes, $29 
for rent, and $7 for a barrel of flour, How much money has 
he left ? 



$l6 



80 TWENTY. 

20. TWENTY. XX. 

SUBTRACTION. 

175. 7. If a lady spends $14 out of a purse containing 
-$60, how many dollars will she still have in her purse ? 

Explanation. — i. Write $60 and under it 
/I the amount spent, or $14. 

2. Since 4 units cannot be taken from 
/ Jl units, take 1 ten from the 6 tens, and change it 

to units, making 10 units ; take the 4 units 
a from the 10 units, leaving 6 units. Write 6 in 

the units' column. 

3. Since 1 ten was taken from the 6 tens, 5 
tens were left ; 1 ten from 5 tens leaves 4 tens. Answer, $46. 

176. Write from dictation and subtract : 

1. 2. 3. 4. 5. 6. 

$90 300 700 800 $60 $50 

23 12 55 38 44 25 

WRITTEN PROBLEMS. 

117. 1. Mr. Brown is now 30 years old. He left school 
14 years ago. How old was he when he left school ? 

2. Henry Arnold has received as a present a book of 90 
pages. He has read all but 18 pages of the book ? How 
many pages has he read ? 

3. Sadie has 28 cents and her brother, Benjamin, has 52 
cents. They buy a box of paints for 65 cents. How much 
money have they left ? 

Jf. Philip had 28 pigeons, and bought 12 more. After 16 
flew away, how many remained ? 

5. From a piece of muslin containing 50 yards, 27 yards 
were cut. How many yards were left ? 



TWENTY. 



81 



2 x 10 = 20 
10 x 2 = 20 


20 -*- 2 = 10 
20 + 10 = 2 


4 x 5 = 20 

5 x 4 = 20 


20 -f- 4 = 5 

20 -*- 5 = 4 



20 TWENTY. XX. 

MULTIPLICATION AND DIVISION. 

178. 1. If 10 cents 
make a dime, how 
many cents are there 
in 2 dimes ? 

2. Add: 

i. 10 + 10. 

2. 5 + 5 + 5 + 5. 

3. 4 + 4 + 4 + 4 + 4. 

4. 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2. 

3. Two 10's are how many ? Ten 2's are how many ? 
Jf, Four 5's are how many ? 
5. 2 times 10 are how many ? 
g. 2 x 10 = ? 

7. 4 x 5 = ? 

8. If candles cost 10 cents a pound, what will 2 pounds cost ? 
5. If five cents make a nickel, how many cents are there 

in 4 nickels ? 

10. If one egg costs 4 cents, what will 4 eggs cost ? 

11. If 20 cents are divided equally between 2 children, how 
much will each child receive ? One half of 20 is how many ? 

12. If 20 cents are divided equally among 4 children, how 
much will each child receive ? One fourth of 20 is how 
many ? 

13. If $20 were divided equally among 5 
men, how much would each man receive ? 
One fifth of 20 is how many ? 

ljf- 4 pansies have 20 petals. How many 
petals has one pansy ? 4x5 = ? 

15. 20-^2= ? 20—10 = ? 

16. 20—4= ? 20-4- 5= ? 

17. How many 2's in 20 ? 4's ? 5's ? 10's ? 



Five 4's are how many ? 
10 times 2 are how many ? 
10 x 2 = ? 
5x4= ? 




82 TWENTY. 

179. Write from dictation and multiply : 

1. 2. 3. 4. 

21 32 44 12 

4 3 2 4 



5. 6. 7. 8. 

34 11 10 11 

_2 _5 _8 7 

180. Write from dictation and divide : 

i. 2. 3. 4. 5. 

5}j55 2)_64 4)_84 7 )_70 4)88 

6. 7. 8. 9. 10. 

6)_66 3)93 4)^48 2 )86 2)88 

WRITTEN PROBLEMS. 

181. i. If one broom costs 32 cents, what will 3 brooms 
cost ? 

2. What is the cost of 4 cans of tomatoes, if one can costs 
12 cents ? 

3. A pound of sweet crackers costs 21 cents. What wilj 
4 pounds cost ? 

Jf. If a board plank is 11 feet long, how long will 5 
such planks be ? 

5. If each of 3 houses contains 13 windows, how many 
windows will the 3 houses together contain ? 

6. In an orchard there are 68 trees ; half of them are 
peach trees. How many peach trees are there in the 
orchard ? 

7. At 8 cents a quart, how many quarts of milk can be 
bought for 88 cents ? 

8. Find one third of 63 ; one fourth of 48 ; and one fifth 
of 55. 



MEASUREMENT. 



VALUES 




10 cents make 1 dime. 



10 dimes 



1 dollar. 



ORAL PROBLEMS. 

182. 1. If a yard of ribbon costs 10 cents, what will half 
a yard cost ? 

2. If a ride on the elevated railroad costs 1 dime, how 
many rides can be had for half a dollar ? 

3. How many dimes in a dollar ? In half a dollar ? 

J/,. If a ride in a horse car costs 5^, how many rides can be 
had for a nickel ? For half a dime ? For 2 dimes ? 

5. How many cents in 4 nickels ? 

6. In 20 cents, how many dimes are there ? In 90 cents ? 

WRITTEN PROBLEMS. 

183. i. If 6 dimes are divided equally among 6 children, 
how many cents will each child receive ? 

$ . If each of 5 little girls has 11 cents, how many cents 
have all together ? 

3. How many dimes will it take to pay 35 cents for a can 
of peaches, 42 cents for a box of soap, 13 cents for sugar ? 



84 MEASUREMENT. 

LENGTHS. 






Long 


Measure 




12 


inches 


make 


1 


foot. 


3 


feet 


Si 


1 


yard. 



ORAL PROBLEMS. 

184. 1. How many inches 
are there in half a foot of 
wire ? In a third of a foot ? 
In a quarter of a foot ? 

2. If a foot of brass chain costs 5 cents, what will a yard 
cost ? 

3. How many yards are there in 6 feet ? In feet ? In 
12 feet ? In 15 feet ? 

4* If 2 yards of rope cost 6 cents, what will 1 foot of 
rope cost ? 

WRITTEN PROBLEMS. 

185. 1. How many inches are there in 4 ft ? In 3 ft. ? 

2. From 4 feet of steel wire 3 inches were cut. How 
much remained ? 

3. In 63 feet, how many yards are there ? 

4. Add 1 ft. 6 in. to 1 ft. 5 in. 

5. From 2 ft. 10 in. take 1 ft. 7 in. 

6. Find how many feet there are in 31 yards. 

7. How many inches are there in a yard ? In a half a yard ? 
In a third of a yard ? In a quarter of a yard ? 

8. Mary gave 5 dimes for half a yard of cloth at 84 cents a 
yard. How much change is due her ? 

9. Frank gave 3 dimes and a nickel for 6 in. of silver wire 
at 62 cents % ft, What is his change ? 



MEASUREMENT. 
LIQUIDS. 



85 






ORAL PROBLEMS. 



Liquid Measure. 



2 pints make 1 quart. 



4 quarts 



1 gallon. 



186. i. If a quart of milk 
costs 8 cents, what will a pint 
cost ? 

2. When milk is selling for 5 
cents a pint, what will a quart of milk cost ? 

3. A quart of cider vinegar may be bought for 5 cents. 
What will be the cost of a gallon at the same price ? 

Jf. If oil is 16 cents a gallon, how much should a quart of 
oil cost ? A pint ? 

5. How many pints in one gallon ? In 2 gallons ? 

6. How many quarts in two gallons ? In 4 gallons ? 

7. How many quarts in 12 pints ? In 18 pints ? 

8. How many gallons in 16 pints ? In 16 quarts ? 

WRITTEN PROBLEMS. 

187. i. In 33 quarts, how many pints are there ? 

2. In 68 pints, how many quarts are there ? 

3. How many gallons are there in 88 pints ? 

4- At 11^ a pint, what will be the cost of a gallon of oil ? 

5. What will be the cost of 2 quarts of olive oil at 21^ a 
pint ? 

6. If 2 quarts of syrup cost 48^, what will a pint cost ? A 
quart ? 3 pints ? 



86 ROMAN NOTATION. 



ROMAN NOTATION. 

188. In the Roman System of Notation, letters are 
employed to represent numbers, viz. : 



Letters. 


I. 


V. 


X. 


L. 


Values. 


1 


5 


io 


50 



189. In combination, their significance is in accordance 
with the following 

PRINCIPLES. 

1. To repeat a letter repeats its value. 
Thus, III. is 3, and XXX. is 30. 

The letters V and L are never repeated. 

II. Wlien one letter follows another of greater value, 
the sum of their values is denoted ; zuhen it precedes, 
the difference between their values is denoted. 

Thus, XXV. is 25, and XL. is 40. 

III. A letter betiveen two of greater value diminishes 
by its value the sum of the values of the other two 
letters. 

Thus, XIX. is 19, and LXIV. is 64. 

190. Express by figures, or the Arabic Notation : 

i. IX. 4. XXXIV. 7. L. io. XLIX. 

2. XXI. 5. XXXIX. s. LXX. n. LXXX. 

3. XXV. e. XLIV. 9. LXIX. 12. LXXXIX. 

191. Express by the Roman Notation : 

1. 14. 3. 32. 5. 55. 7. 68. 9. 79. n. 54. 

2. 23. 4. 40. 6. 44. s. 81. io. 38. 12. 89. 



SECTION III. 



NOTATION AND NUMERATION. 

1. How many tens are represented by each of the following ? 

10 20 30 40 50 60 70 80 90 

2. Write the words for which the preceding figures stand. 



make 



Ten Tens 



make 



One Hundred. 



192. i. Ten tens, or One Hundred units, are repre- 
sented, not by two figures, but by writing 10 at the left 
of 0, or 1 at the left of two ciphers ; thus, 100. 

2. In the same manner : 

Twenty tens, or two hundred units, are written 200. 

Thirty tens, " three hundred units, 

Forty tens, " four hundred units, 

Fifty tens, " five hundred units, 

Sixty tens, " six hundred units, 

Seventy tens, " seven hundred units, 

Eighty tens, " eight hundred units, 

Ninety tens, " nine hundred units, 



a 


a 


300 


a 


£ £ 


400 


a 


££ 


500 


a 


££ 


600 


a 


££ 


700 


a 


££ 


800 


i( 


££ 


900 



8g NOTATION AND NUMERATION 

HUNDREDS, TENS, AND UNITS. 

1. How many tens and units are represented by each of 
the following numbers ? 

100, 101, 110, 111, 116, 123, 175, 188, 199. 

2. For what does the left-hand figure stand in each of these 
numbers ? Is it in the first, second, or third place ? 

3. What does the figure in the first, or right-hand, place 
represent ; units, tens, or hundreds ? In the second place ? 
In the third, or left-hand place ? 

Jf. In order to represent two hundred thirty-five, where 
should the 2 be written ? The 3 ? The 5 ? Why ? 

193. Numbers composed of hundreds, tens, and units are 
expressed by writing the figure representing the tens at the 
right of the figure representing the hundreds, and the figure 
representing the units at the right of the figure representing 
the tens. Thus, 

2 hundreds, 3 tens, and 4 units, or 

two hundred thirty-four, are written 234. 
4 hundreds, 5 tens, and 6 units, or 

four hundred fifty-six, " "' 456. 

9 hundreds, 9 tens, and 9 units, or 

nine hundred ninety-nine, " " 999. 

194. Write in figures the folloiving numbers : 

1. One hundred nine. 5. Four hundred seventy. 

2. Three hundred thirty-seven. 6. Six hundred six. 

3. Five hundred fifty. 7. Seven hundred four. 
Jf. Two hundred. 8. Nine hundred ninety. 

195. Ten hundreds, or one hundred tens, or one thou- 
sand units, are expressed by writing 1 with three ciphers 
at the right ; thus, 1000. 



NOTATION AND NUMERATION 89 

196. Numbers of more than three figures are divided into 
periods of three figures each, commencing at the right. 



Names of 


2d. 




ls£. 


Periods. 


Thousands* 




Units, 


Names of 
Places. 


Hundreds "| 
Tens 
Units J 


02 


Tens 
Units J 


Number. 


3 4 5, 


7 


2 1 


Orders. 


rd rd rd 
-+2 -+=> -+2 

CO o ^ 


CO 





197. i. In each period the names of the three places are 
units, tens, and hundreds. 

2. Each period is read as if it formed a number by itself, 
and then the name of the period is read. 

3. The preceding number is read three hundred forty-five 
thousand, seven hundred twenty-one. 

Note. — Observe that in reading any given whole number the final 
letter, s, is omitted in naming the period ; units' period is not named ; 
and the word and is not used. 

198. Head the following numbers : 

i. 23,458. 4. 300,707. 7. 80,008. 

s. 125,500. 5. 400,040. 8. 909,990. 

3. 203,660. 6. 500,005. 9. 999,999. 

199. Empress the following numbers in figures : 

1. Fifty thousand, five hundred. 

2. One hundred thousand, three hundred forty. 

3. Two hundred two thousand, two. 

4. Four hundred thousand, four hundred. 

5. Nine hundred ninety thousand, thirty-seven. 

6. Nine hundred nine thousand, nine hundred ninety. 



90 ADDITION. 

ADDITION. 

TWENTY TO THIRTY. 

200. 1. If a sketch book costs 19 cents, and a lead pen- 
cil 1 cent, what will the two together cost ? 

2. If a pound of raisins is worth 18 cents and a quarter 
of a pound of nuts 2 cents, what are both worth ? 

3. Nettie paid 17 cents for a yard of lace and 3 cents for 
some needles. What did she pay for both ? 

Jf. What was the entire cost of a quire of paper at 16 cents 
and 2 postage stamps at 2 cents each ? 

5. Mr. Cobb paid 15 cents for a water-melon and 5 cents 
for a musk-melon. What did he pay for both ? 

6. How much money does Percy spend, if he buys 7 post- 
age stamps at 2 cents each, and a package of envelopes for 
6 cents ? 

7. If Howard buys 4 oranges at 3 cents each, and 4 
apples at 2 cents each, how much money will he have to pay ? 

8. Kate's drawing book cost 11 cents and her 3 pencils, 
3 cents each. What was the cost altogether ? 

9. Madge is 19 years old. How old will she be in 2 years ? 

10. What will be the cost altogether of 2 quarts of black-, 
berries at 9 cents a quart and an orange for 3 cents ? 

11. One piece of muslin contains 17 yards ; another piece 
contains 4 yards more than the first piece. How many yards 
are there in the second piece ? 

12. What fare should a street-car conductor charge for 
3 men at 5 cents each, and 2 children at 3 cents each ? 

13. If a yard of ribbon costs 14 cents, how much will a 
yard and a half cost ? 

1]},. In one bunch there are 13 grapes; in another, 8. How 
many grapes are there in the two bunches ? 

15. How much money has Hugh, if he has in one pocket 2 
five-cent pieces and one cent, and in another pocket, a dime ? 




TWENTY TO THIBTY. 91 

16. Mr. Pierce received 19 letters and 3 telegrams in one 
day. How many of both did he receive ? 

17. There are 17 sparrows on one telegraph wire and 5 o 
another. How many sparrows are there on both wires ? 

18. Bertha bought 2 spools of silk thread at 8 cents eacn 
and 2 pieces of tape at 3 cents each. How much altogether 
did they cost her ? 

19. If a third of a yard of ribbon costs 5 cents, what will 
a yard of the ribbon and 7 cents' worth of buttons cost ? 

20. In making a journey, a party 
traveled 14 miles by sleigh and 8 miles 
by railroad. How far did they travel ? 

21. What will a rocking-chair, $13, 
and a table, $9, together cost ? 

22. A builder built 12 houses in one block and 10 houses in 
another. How many houses did he build in the two blocks ? 

23. A row of maple trees contains 4 more trees than a row 
of poplar trees. If there are 19 poplar trees in the row, how 
many maple trees are there ? 

24- In one box there are 17 pounds of tea ; another box 
contains 6 pounds more. How many pounds of tea are there 
in the second box ? 

25. Emma read one summer 19 books ; Maggie read 5 
more than Emma. How many books did Maggie read ? 

26. Mrs. Cary's bonnet cost $18 ; her shawl cost $6 more 
than the bonnet. How much did the shawl cost ? 

27. If lace trimming is sold for 17 cents, or 7 cents less 
than the cost, what was the cost ? 

28. One slate cost 4 cents ; another slate cost 4 times as 
much ; a third slate cost 8 cents more than the second slate. 
How much did the third slate cost ? 

29. Philip bought 3 rubber balls at 5 cents each and 3 
more at 3 cents each. How much did the 6 balls cost him ? 

30. What will be the cost altogether of 2 quarts of berries 
at 7 cents a quart and 2 quarts at 5 cents a quart ? 



92 ADDITION. 

31. At a book store George bought a First Reader for 15 
cents, a lead pencil for 4 cents, and a slate for 6 cents. 
How much did he pay for all ? 

32. Hannah spent 10 cents for ribbon, 8 cents for tape, 
and 7 cents for pins. How much did she spend ? 

33. Mr. James ordered from a store celery, 10 cents; berries, 
7 cents ; and lettuce, 8 cents. How much did he have to pay? 

34- Willie bought a top for 6 cents, a ball for 10 cents, and 
some marbles for 9 cents. How much money did he spend ? 

35. Mamie had 3 five-cent pieces, and Sarah had 2 five-cent 
pieces. How much money did they both have ? 

36. In 3 collar boxes there are 10, 9, and 7 linen collars. 
How many collars altogether are there ? 

201. Announce sums at sight: 

Read answers: 1. From left to right ■; 2. From right to left ; 3. From 
top to bottom; 4. From bottom to top; 5. As directed by the teacher. 



1st. 


m. 


3d. 


4th. 


5th. 


6th 


7th. 


8th. 


9th. 


1 


3 


3 


5 


5 


7 


7 


9 


9 


19 


18 


17 


16 


15 


14 


13 


12 


11 





S 2 


2 


4 


4 


6 


6 


8 


8 


10 


B 


hi 


18 


17 


16 


15 


14 


13 


12 


11 




s 3 


5 


5 


7 


7 


9 


9 


9 


• 8 


C 


m 


18 


17 


16 


15 


14 


13 


15 


16 




i i 


4 


6 


6 


8 


8 


10 


10 


7 


I> 


ju 


18 


17 


16 


15 


14 


13 


14 


17 




i 6 


8 


8 


9 


8 


10 


10 


10 


6 


E 


1» 


18 


17 


16 


19 


18 


17 


16 


18 




I 7 


7 


9 


10 


9 


9 


10 


5 


10 


F 


» 


18 


17 


15 


19 


18 


19 


19 


10 



TWENTY TO THIRTY. 93 

202. What is the value of: 

lt 5 X 4 + 8 — 3 + 4—2? 4. 20-T-2 + 10+ 5+ 4—8? 

2. 3 x 6 + 3 + 6 + 2-l? 5. 18-^9 + 9 + 10+ 1-2? 

3. 6x2 + 9 + 6-7 + 4? 6. 16H-4+ 7+ 9+ 6 + 3? 

203. Find the sums: 

1. 2. 3. 4. 5. 6. 



2 


2 


5 


3 


2 


2 


5 


4 


1 


2 


3 


1 


1 


8 


5 


4 


4 


7 


3 


2 


3 


4 


5 


1 


8 


3 


5 


6 


4 


4 


5 


4 


2 


4 


6 


7 



204'. Write from dictation and find the sums : 

1. Explanation. — 1. Write the num- 

/ *? D /? ^ ers so ^ na ^ un ^ s °^ the same order 

^) stand in the same column. 

2 024 2 < Adding the units, the result is 

/ / / O) 20 units or 2 tens and units. Write 

7 y U J^iJ under the units. Carry the 2 tens 

9) / 7* JR ^° ^ ne c °i umn °f tens. 

__^j> ___ & Adding the 2 tens to the tens' 

// /I y ^T // J column, the result is 15 tens or 1 hun- 

4 U y J DU Ans. dred and 5 tens< Write 5 under the 

tens and carry the 1 hundred to hun- 
dreds' column. 

4. Adding the 1 hundred to the hundreds' column, the result is 11 
hundreds or 1 thousand and 1 hundred. Write 1 under the hundreds 
and carry the 1 thousand to thousands' column. 

5. Adding the 1 thousand to the thousands' column, the result is 10 
thousands or 1 ten-thousand and thousands. Write under the 
thousands and 1 in the column of ten-thousands. Answer, 1 ten- 
thousand, thousands, 1 hundred, 5 tens, units, or 10,150, 



94 





ADDITION. 




2. 


3. 


4. 


5. 


2,398 


1,099 


897 


1,030 


1,079 


2,069 


1,729 


2,279 


3,404 


1,140 


2,132 


' 3,003 


2,920 


1,312 


4,052 


1,099 


6. 


7. 


8. 


9. 


663 


79 


2,026 


683 


1,087 


2,790 


1,395 


2,437 


904 


1,044 


880 


1,086 


2,257 


3,037 


2,109 


2,014 




WRITTEN PROBLEMS. 

205. 1. A merchant ships to Europe 1,024 bushels of 
wheat, 2,038 bushels of corn, 967 
bushels of oats, and 3,102 bushels of 
barley. How many bushels of grain 
does he ship ? 

2. In a village there are 999 men, 
1,364 women, 1,101 boys, and 1,467 
girls. How many people are there in 
the village ? 

3. A farmer has 1,625 acres of corn, 666 of rye, 330 of 
tobacco, 126 of flax, and 1,494 of wheat. How many acres 
in all has he ? 

4- In a bank there are $1,569 in gold, $2,562 in silver, 
$1,329 in bills, and $549 in small currency. How much 
money is there in the bank ? 

5. In a large store-house there are 1,945 barrels of flour, 
2,075 barrels of potatoes, 1,648 barrels of sugar, and 2,348 
barrels of apples. How many barrels altogether are there 
in the store-house ? 

6 . There are in a state 1,049 miles of rivers, 2,738 miles 
of railroads, 6,189 miles of wagon roads, and 374 miles of 
canal. How many miles of road way are there ? 



TWENTY TO THIRTY. 95 

7. A man bought four farms for $1,097 ; $3,129 ; $2,096 ; 
$1,085. How much did they cost him altogether ? 

8. Four houses were sold for $2,390; $2,760; $2,860; 
$1,990. For how much were the four houses together 
sold ? 

9. Mr. Hope bought a house and lot for $4,190 ; a farm 
for $2,880; some cattle for $1,960; and some horses for 
$875. How much did he pay for all ? 

10. In a forest there were 568 elm trees, 1,059 beech 
trees, 2,846 maple trees, and 377 spruce trees. How many 
of these trees in all were there ? 

11. In a fish market there were 302 trout, 1,687 white fish, 
1,578 blue fish, 653 salmon, and 1,779 bass. How many fish 
were there in the market ? 

12. Mr. Griggs had $2,405 in gold, $1,875 in silver, $3,197 
in paper money, $655 in five-cent pieces, and $147 in copper 
coins. How much money did he have ? 

13. After a battle it was found that 483 men had been 
killed, 1,229 men had been wounded, 678 men had deserted, 
or run away, and 2,697 men remained unhurt. How many 
men altogether were there before the battle ? 

IJf.. A ship carried 2,031 barrels of flour, 1,707 barrels of 
sugar, 2,659 barrels of salt, 1,078 barrels of potatoes, and 594 
barrels of apples. How many barrels altogether did the ship 
carry ? 

15. An army consisted of 41,987 infantry, 15,684 artillery, 
and 12,496 cavalry. How many men were there in the army ? 

16. Find the united population of the four following cities 
in the State of New York : Buffalo, 155,134 ; Albany, 90,758 ; 
Rochester, 89,366 ; and Syracuse, 51,792. 

17. What is the total cost of four houses costing respect- 
ively : $5,685, $6,975, $4,890, and $7,345 ? 

18. In five different years a merchant purchased the fol- 
lowing bills of goods : $8,368, $7,845, $6,579, $4,526, and 
$2,371. What was the entire amount of his purchases ? 



96 SUBTRACTION. 



SUBTRACTION. 

TWENTY TO THIRTY. 

206. i. On a stand there were 21 bananas in a bunch. 
After 1 had been sold, how many were left ? After 2 had 
been sold ? 3 ? 4 ? 

2. If Matthew shoots 3 times 7 quails and Henry shoots 
5 less than Matthew, how many does Henry shoot ? If he 
shoots 6 less than Matthew, how many does he shoot ? 
7 less ? 

3. Mary having 2 dimes and 1 cent, had how much left 
after spending 8 cents ? 9 cents ? 10 cents ? 

4* A farmer owning a flock of 22 sheep sold first 2 sheep, 
then a third sheep, and then a fourth sheep. How many 
sheep did he have left each time ? 

5. Edith is now 22 years old. How old was she five years 
ago ? 6 years ? 7 years ? 

6. Anna had 4 five-cent pieces and a two-cent piece. 
How much did she have after spending 8 cents ? 9 cents ? 
10 cents ? 

7 . How many yards remained in a piece of muslin con- 
taining 23 yards after cutting off 3 yards ? 4 yards ? 5 
yards ? 6 yards ? 

8. A suit of clothes cost $23. How much did the coat 
and vest cost, if the trousers cost $7 ? $8 ? $9 ? $10 ? 

9. How much change must Mary receive on paying for a 
five-cent spool of thread with a quarter of a dollar ? 

10. From a bolt of cloth containing 24 yards, 6 yards were 
cut. How many yards were left ? 

11. Some ribbon and some lace together cost 25 cents. 
The ribbon cost 6 cents. How much did the lace cost ? 

12. A grocer buys 13 barrels of flour and then 9 barrels. 
After selling 4 barrels and then 3 barrels, how much flour 
does he still have ? 



TWENTY TO THIRTY. 97 

207. Announce differences at sight: 

Read answers : 1. From left to right ; 2. From right to left ; 3. From 
top to bottom ; 4. From bottom to top ; 5. As directed by the teacher. 

1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 
(21 21 21 21 21 21 21 21 21 
A j 1 3 5 8 10 7 64 2 



(21 22 22 22 22 22 22 22 22 
B ' \J. _? _i J? A A 12 Jt _? 



(22 23 23 23 23 23 23 23 23 

C j 7 3 10 4 9 5 86 7 



(24 24 24 24 24 24 24 25 25 
D ) 4 6 9 10 8 7 55 6 



j 25 25 25 25 26 26 26 26 26 
E j 7 9 8 10 6 9 78 10 



(27 27 27 27 28 28 28 29 29 
' 10 9 8 8 10 9 9 10 



u 



208. Find the value of: 

i. 2x4 + 10 + 5—4 + 2 — 7. 9. 3x5 + 10-8 + 9 — 7 + 8. 

2 . 5x3 + 8—7 + 5-8-4. io. 4x5 + 7-9 + 5-8-7. 

s. 3x3 + 9 + 4—6 + 7—5. n. 5x4 + 8—10 — 9 + 3—6. 

4 . 2x6 + 9—6 + 7—8—5. 12.12-4-6 + 8 + 7 + 8 — 9 + 7. 

s. 12-4-3 + 9—4 + 8 + 5—9. is. 14-4-7 + 9 + 9—2 + 6—9. 

o. 15-4-5 + 8—4 + 10 + 6 — 7. i*. 16-4-4 + 8 + 8 + 6—8—9. 

7 . 18^-3 + 6 + 9 — 6 + 8 — 6. is, 18-4-2 + 7 + 9 — 7—10 + 6. 

*. 18-5-9 + 9 + 9—7 + 8—4. 16. 20-4-5 + 9 + 8 + 6 — 9—10. 



6,5<)4 



98 SUBTRACTION. 

309. Write from dictation and find the differences : 

1 - Explanation. — 1. Write the larger 

Q JiQQ number first ; under it write the 

I,* smaller number, placing units under 

units, tens under tens, etc. 

2. Since there are neither units nor 

G) O /I /) a tens in the larger number from which 

^jyOU^ A.US. tQ take the unitg and teng of the 

smaller number, 1 hundred is taken 
from the 4 hundreds, leaving 3 hundreds; the 1 hundred is then 
changed to tens, making 10 tens; 1 of these tens is taken from the 10 
tens, leaving 9 tens; this 1 ten is then changed to units, making 10 
units. We then have 3 hundreds, 9 tens, and 10 units' equal to 400. 

3. 1 unit from 10 units leaves 9 units. Write 9 under the units. 
9 tens from 9 tens leave tens. Write under the tens. 

4- Since 5 hundreds cannot be taken from 3 hundreds, 1 thousand is 
taken from the 9 thousands, leaving 8 thousands ; it is then changed to 
hundreds, making 10 hundreds, which added to the 3 hundreds make 
13 hundreds ; 5 hundreds from 13 hundreds leave 8 hundreds. Write 
8 under the hundreds. 

5. 6 thousands from 8 thousands leave 2 thousands. Write 2 under 
the thousands. Answer, 2,809. 

Proof. — Add the answer to the smaller number, and if the result is 
equal to the larger number, the work is correct. 



a. 


3. 


4. 


5. 


3,400 


2,300 


6,400 


2,506 


2,905 


1,409 


1,508 


1,308 


6. 


7. 


8. 


9. 


4,000 


4,246 


7,355 


2,001 


2,678 


3,167 


6,467 


1,998 




WRITTEN 


PROBLEMS. 





210. 1. A house and lot were bought for $3,540, and 
sold for $2,655. How much was lost by the sale ? 

2. If a man pays $5000 for a farm that he afterwards sells 
for $3510, how much does he lose ? 



TWENTY TO THIRTY, 



99 




3. If a merchant sells for $8660 a vessel that he bought for 
$6000, how much does he make ? 

4. The summit of Mt. Washington is 6428 ft. high. The 
summit of Mt. Saddleback is 4000 ft. high. Which is the 
higher, and how much ? 

5. The Ohio Kiver is 948 miles long, and the Missouri 
River is 4350 miles long. Which is the longer, and how much ? 

6. Mr. Jones bought his grocery 
for $7500; he sold it for $9350. 
Did he make or lose, and how 
much ? 

7. In 874 the Northmen settled 
in Iceland. How many years ago 
was it ? 

8. How many years is it since 

America was discovered by Columbus in the year 1492 ? 

9. The first book was printed in the year 1423 ; the first 
newspaper was printed in the year 1615. How many years 
were there between the printing of the first book and the 
printing of the first newspaper ? 

10. George Washington was born in the year 1732 and died 
in 1799. How many years did he live ? 

11. In 1880 the number of people in Philadelphia was 
847,000; in Brooklyn, 567,000. Which city contained more 
people than the other, and how many more ? 

12. The number of horses in New York in 1875, was 
650,200. Five years later the number was 610,000. Did the 
number become greater or less, and how much ? 

13. A man bought a house and lot for $8,695 ; he sold it 
for $9,760. Did he make or lose, and how much ? 

14- The number of passengers on a line of street-cars 
was 3,450 one day ; 2,790 the next day. How much greater 
or less was the number on the second day? 

15. A company bought v goods for $4,698 and sold them 
for $5,733. How much did the company make or lose ? 



100 



MULTIPLICATION AND DIVISION. 



MULTIPLICATION AND DIVISION. 



TWENTY TO THIRTY. 



211. 1. How many coun- 
ters will there be in 3 rows, 
if each row contains 7 ? 

2. How many counters 
will there be in 7 rows, if 
each row contains 3 ? 

3. Divide 21 counters 
into rows of 7 counters 
each. How many rows are 
there ? 



4. Divide 21 



counters 
counters 



3 x 


7 = 21 


21- 


3 = 


7 


7 x 


3 = 21 


21- 


7 = 


3 


3 x 


8 = 24 


24^- 


3 = 


8 


8 x 


3 = 24 


24-r- 


8 = 


3 


3 x 


9 = 27 


27-^- 


3 = 


9 


9 x 


3 = 27 


27-*- 


9 = 


3 


3 x 


10 = 30 


30 -^ 


3 = 


10 


lO x 


3 = 30 


30-f- 


10 = 


3 



into rows of 3 

each. How many rows are there ? 21-r-3=? 21 —-7=? 

5. In each of 3 boxes there are 8 spools of cotton. How 
many spools altogether are there ? 

6. On each of 8 plates there are 3 peaches. How many 
peaches are there on the 8 plates together ? 

7. If $24 are divided equally among 3 men, how many dol- 
lars will each man receive ? 

8. A boy bought 8 apples for 24 cents. How much did 
he pay for each apple ? 

9. How much does Mr. Thompson earn in 3 weeks at $9 
per week ? 

10. 27 cents will buy how many lead pencils at 3 cents 
each ? How many paper cutters, at 9 cents each ? 

11. At 10 cents ekch, what will 3 boxes of paints cost ? 

12. At 3 cents each, what will 10 oranges cost ? 

13. A jeweler sold 10 clocks for $30. How much did he 
receive for each clock ? 

14- If one man can do a piece of work in 30 days, how 
long will it take 3 men to do the same work ? 



TWEN1Y TO THIRTY, 101 

212. State products at sight : 

Read answers : 1. From left to right ; 2. From right to left; 3. From 
top to bottom ; 4. From bottom to top ; 5. As directed by the teacher. 

1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 
(8 3 9 3 83 10 3 4 

A j 3 7 39 38 3 10 4 



5 


4 


6 


3 


4 


3 


5 


3 


3 


4 


5 


3 


6 


3 


4 


3 


5 


3 


5 


2 


6 


2 


9 


2 


4 


10 


2 


2 


5 


2 


6 


2 


9 


2 


2 


10 



c 



213. State quotients at sight: 

Read answers : 1. From left to right ; 2. From right to left ; 3. From 
top to bottom ; 4. From bottom to top ; 5. As directed by the teacher. 

1st. 2d. 3d. Jfth. 5th. 6th. 7th. 8th. 9th. 

A {3)21 7)21 3)24 8)24 3)27 9)27 3)30 10)30 4)16 

B j 4)20 5)20 2)18 9)18 3)12 4)12 3)15 5)15 3)9_ 

C j 2)6_ 3)6_ 2)10 5)10 2)12 6)12 2)20 10)20 2)8_ 

214. Co2>?/ tfwd! multiply : 

l. Explanation. — 1. Three 5's are 15 or 

ry / p- 1 ten and 5 units. Write the five in units' 

' 4^ *J place. 

Q 2. 3 times 4 tens are 12 tens ; which 

with the 1 ten from the units' product 

(7) (J) (7) pr a make 13 tens or 1 hundred and 3 tens. 

& } & O O A.US. Write the 3 in tens' place. 

3. 3 times 7 hundreds equals 21 hun- 
dreds, which with the 1 hundred from the tens' product make 22 hun- 
dreds or 2 thousand 2 hundred. Write the 2 hundreds in hundreds' 
place and the 2 thousands in thousands' place. Answer, 2,235. 



102 MULTIPLICATION AND DIVISION 

2. 3. 4. 5. 6. 

345 596 524 978 759 

_i _? _i 1 _ 3 

7. S. 9. 10. 11. 

213 132 123 654 432 
8 _9 _J_ _3 _5 

215. Copy and divide : 

I* Explanation. — 1. 3 is contained in 

7 hundreds 2 hundred times with 1 hun- 
dred remaining. Write the 2 hundred 
in hundreds' place. 
f£ D / AtIS. ®- The 1 hundred, remainder, is equal 

to 10 tens, which with the 6 tens make 
16 tens. 3 is contained in 16 tens 5 tens 
times with 1 ten remaining. Write the 5 tens in tens' place. 

3. The 1 ten, remainder, equals 10 units, which with the 2 units make 
12 units. 3 is contained in 12 units 4 times. Write the 4 in units' place. 
Answer, 254. 



3)762 



2. 

3)582 


3. 

4)456 


4. 

2)996 


5. 
3)255 


6. 

3)891 


7. 

8)256 


8. 

9)288 


9. 

7)224 


10. 

6)192 


11. 

5)655 




WRITTEN PROBLEMS. 





216. i. Three men bought equal parts of a farm costing 
How much did each man pay ? 

2. If one horse costs $685, what will 3 such horses cost ? 

3. A barrel of flour weighs 196 pounds. How much do 3 
barrels weigh ? 

4- If a quantity of oil costs $665, what will one fifth of it 
cost ? 

3. One third of a quantity of salt cost $168. What did it 
all cost ? 



TWENTY TO THIRTY. 103 

6. If 8 men together earn $976, what does each man earn ? 

7. If a boy can earn $13 per month, how much can he 
earn in 9 months ? 

8. 7 suits of clothes of equal value cost $224. What was 
the cost of one suit ? 

9. If 2 men make equal parts of $976, what does each man 
make ? 

10. 3 sisters together owned a city lot; each sister's share 
was worth $275. What was the value of the entire lot ? 

11. How much will Fred save in 6 months, if he saves $32 
per month ? 

12. If 9 book-cases are sold for $288, what is the price of 
one of the book-cases ? 

13. If each of 13 houses costs $7,826, what will the 13 
houses together cost ? 

ljj,. 3 regiments each contained 875 men ; what was the 
entire number of soldiers in the 3 regiments ? 

15. & railroad company built a railroad 3 miles long, each 
mile costing $58,760. How much did the 3 miles cost ? 

16. A builder borrowed $8,950 on each of 32 houses. 
How much money did he borrow ? 

17. A merchant made $323 in each of 29 months. How 
much did he make in the entire time ? 

18. A grocer made $264 in 8 months ; how much did he 
make per month ? 

19. If 9 houses cost $19,179, how much will each house 
cost ? 

20. In 7 tanks there are 217 gallons of oil ; how many 
gallons are there in each tank ? 

21. If there are 63 gallons of water in one hogshead, how 
many gallons are there in 23 hogsheads ? 

22. A man left a fortune of $154,714 to be divided equally 
among 7 children ; how much did each child receive ? 

23. What will be the cost of a farm containing 32 acres at 
per acre ? 




104 ADDITION. 



ADDITION. 

TWENTY TO FORTY. 

217. i. A dozen buttons cost 29 cents, and a skein of 
worsted 1 cent. What was the cost of both ? 

2. Margaret pays 28 cents for a pine- 
apple and 2 cents for a bunch of grapes. 
What does she pay for both ? 

3. A man lives on a farm containing 
27 acres. If he buys 3 acres more, how 
many acres will the farm contain ? 

J/,. Harriet paid 26 cents for a yard of cloth and 2 cents 
each for 2 knitting needles. How much did she pay for all ? 

5. If a peck of apples costs 25 cents and 5 pears cost 1 cent 
each, what will they cost altogether ? 

6. Edgar gave 24 cents for a blank book and 6 cents for 
a slate. How much did he give for both ? 

7. A ship sailed 23 miles one hour and 7 miles the next 
hour. How far did it sail in the two hours ? 

8. A peck of sweet potatoes cost 22 cents and 2 boxes of 
berries cost 4 cents a box. What was the entire cost ? 

9. Of two flag-poles, one was 21 feet long ; the other was 
9 feet longer. How long was the second pole ? 

10. Each of 2 cash boxes contains $10 and each of 2 
other boxes, $5. How much money is there in the 4 cash 
boxes ? 

11. Wheaton is 29 years old. How old will he be in 2 years ? 

12. 2 yards of calico cost 28 cents and some thread cost 
3 cents. What was the entire cost ? 

13. If the first piece of muslin contained 27 yards, and 
the second piece, 4 yards more than the first, how many 
yards would the second piece contain ? 

14- A man bought 24 pounds of sugar and 7 pounds of 
coffee. How many pounds altogether did he buy ? 



TWENTY TO FOBTY. 105 

15. If in one pocket Hugh had 4 five-cent pieces and 1 
cent, and in another pocket he had 5 two-cent pieces, how 
much money did he have altogether ? 

16. In one letter-box there were 29 letters ; in another, 
3 letters. How many were there in both boxes ? 

17. On a sail-boat there are 28 men. In each of 2 row- 
boats there are 2 men. How many men are there altogether ? 

18. If there are 27 roses on one bush and 5 on another, 
how many roses are there on the two bushes ? 

19. Mr. Shaw earns $26 in a week and his son earns $6. 
How much do they both earn ? 

20. How much coal did a man put into his cellar, if he 
put in 25 tons of nut coal and 7 tons of stove coal ? 

21. Sarah saved 24 cents one week and 8 cents, the next 
week. How much did she save in the two weeks ? 

22. A farmer has an orchard con- 
taining 23 apple trees and 9 pear trees. 
How many trees of both kinds has he ? 

23. On the shore there were 22 
Indians and on the river there were 5 
canoes, each containing 2 Indians. 
How many Indians in all were there ? 

24- John has 2 dimes and 3 three-cent pieces. Henry has 
4 cents more than John. How many cents has Henry ? 

25. On a beach there are 29 children. The number of 
ladies there is 5 greater than the number of children. How 
many ladies are there on the beach ? 

26. A merchant bought a cloak for $28, and sold it so as 
to gain $6. For how much did he sell it ? 

27. One house is 26 feet wide ; another house is 8 feet 
wider. How wide is the second house ? 

28. Carl paid 25 cents for a base-ball and 9 cents for candy. 
How much did he spend ? 

29. Anna has 10 cents more than Job, who has 2 dimes 
and 2 two-cent pieces. How much money has Anna ? 




106 ADDITION. 

30. Mrs. Kane bought a pitcher for 21 cents, a tumbler for 
8 cents, and a salt-cellar for 6 cents. What did she pay for 
the three articles ? 

31. Bertha bought a box of starch for 28 cents and a bar 
of soap for 7 cents. What did both cost her ? 

32. Maud purchased some oranges, 14 cents ; chestnuts, G 
cents ; plums, 7 cents ; and walnuts, 8 cents. What sum of 
money did she spend ? 

33. A shoe dealer sold 7 pairs of boots, 9 pairs of slippers, 
10 pairs of overshoes, and 9 pairs of children's shoes. How 
many pairs altogether did he sell ? 

34- How many yards of muslin were there in a piece from 
which after 7 yards had been cut, 29 yards remained ? 

218. Announce sums at sight: 

Read answers : 1. From left to right ; 2. From right to left ; 3. From 
top to bottom ; 4- From bottom to top ; 5. As directed by the teacher. 



1st. 


2d. 


3d. 


4th. 


5th. 


6th. 


7th. 8th. 


9th. 


1 


3 


3 


5 


5 


7 


7 9 


9 


39 


38 


37 


36 


35 


34 


33 33 


31 



c 



D 



E 



F 





















3 


3 


4 


4 


6 


6 


8 


8 


10 


39 


38 


37 


36 


35 


34 


33 


33 


20 


3 


5 


5 


7 


7 


9 


9 


9 


10 


29 


38 


37 


36 


25 


34 


33 


35 


33 


4 


4 


6 


6 


8 


8 


10 


10 


10 


39 


38 


37 


36 


35 


34 


39 


34 


33 


6 


8 


8 


9 


8 


10 


10 


10 


8 


39 


38 


37 


36 


39 


38 


37 


36 


36 


7 


7 


9 


10 


9 


9 


10 


10 


8 


39 


38 


37 


35 


39 


38 


39 


31 


21 



TWENTY TO FORTY. 107 

219. Find the value of: 



1. 


3x 


6— 8 + 9 + 9 + 7. 


2. 


4x 


4+ 4—2 + 9 + 8. 


3. 


5x 


2 + 10 + 9 + 8—5. 


4. 


6x 


3- 7 + 9 + 7 + 7. 


5. 


16-s- 


8+ 9 + 9 + 8 + 9. 


6. 


20-*- 


10 + 10 + 8 + 6 + 7. 


7. 


12-r- 


4+ 7 + 8-5 + 9. 



9. 


9-J-3 + 7 + 


8- 2 + 7. 


10. 


8-^-2+ 9 + 


9 + 10 — 2. 


11. 


7x2 + 10- 


1+ 9 + 5. 


12. 


3x6— 3 + 


9+ 6 + 7. 


13. 


2x5+ 9+ 


6+ 8 — 2. 


14. 


4x6+ 5 + 


8+ 2-9. 


IS. 


8x4+ 7- 


9— 6 + 8. 



8. 15-f- 3+ 9 + 9 — 3 + 10. ml2-i-4+ 8 + 10+ 9 + 5. 

220. Find the sums : 



1. 


2. 


3. 


4. 


5. 


6. 


7. 





4 


4 


1 


3 


8 


4 


6 


5 


3 


2 


1 


4 


2 


2 


2 


4 


4 





3 


1 


3 


3 


2 


3 


2 


1 


2 


4 


4 


4 


9 


8 


2 


3 


3 


6 


3 


2 


2 


1 


2 


8 


3 


1 


3 


8 


9 


3 


5 


2 


6 


2 


2 


3 


8 


2 


4 


2 


3 


4 


1 


2 


3 


2 


7 


6 


2 


6 


3 


2 


4 


2 


3 


5 


1 


7 



221. Write from dictation and find the sums , 



1. 


2. 


3. 


4. 


5. 


$257 


$1652 


$344 


$1215 


$1205 


1764 


473 


1556 


986 


894 


445 


2648 


743 


1124 


2178 


2562 


337 


1378 


883 


769 


724 


1872 


747 


1379 


1543 


1463 


548 


1652 


563 


1677 


156 


1217 


125 


1326 


121 



108 







ADDITION. 






6. 


7. 


8. 


9. 


10. 


1625 


723 


757 


248 


1431 


717 


1919 


1870 


1701 


457 


1974 


867 


949 


799 


1078 


868 


1078 


2167 


1320 


989 


2087 


690 


500 


487 


1640 


1065 


1289 


1088 


1676 


1838 


212 


2001 


256 


1520 


402 




WRITTEN PROBLEMS. 

222. i. The numbers of soldiers in four regiments were 
respectively : 839 ; 908 ; 785 ; and 798. How many soldiers 
altogether were there ? 

2. In a battle the numbers of 
Indians of different tribes were as 
follows : 569 ; 887 ; 806 ; 784 ; and 
898. How many Indians in all 
were there ? 

3. In six months, a merchant's 
sales amounted to $2007 ; $1658 ; $989 ; $1600 ; 
and $1756. What was the amount for the six months to- 
gether ? 

4. A builder paid out for stone, $1989 ; for bricks, $899 ; 
for lumber, $1278 ; and for labor, $1876. How much money 
did he pay out ? 

5. Ships carried in one month from New York the follow- 
ing numbers of bales of cotton: 1,875; 967; 2,908; 898; and 
1,064. How many bales altogether were shipped ? 

6. The numbers of visitors to Coney Island in one day 
were: 1,689 men; 3,998 women; 2,067 girls; and 1,580 boys. 
What was the total number of visitors ? 

7. On five different days the sums of money forwarded by 
an agent were: $2,058; $1,875; $997; $1,886; and $1,569. 
How much money did he forward ? 



TWENTY TO FORTY. 109 

8. In making four long journeys, a man traveled the fol- 
lowing numbers of miles: 1,876; 2,488; 1,853; and 1,977. 
How far did he travel ? 

9. A man deposited in five different banks the following 
amounts of money : $2,050; $1,878; $999; $1,485; and 
$1,639. How much money did the man have in these 
banks ? 

10. Mr. Kelley's expenses for a year were : $1,875, rent; 
$1,549, horses; $1,268, table; $1,447, help; and $1,660, 
traveling and other expenses. What was the amount of his 
expenses for the year ? 

11. What will be the united weight of 1 sack of barley 
containing 2 bushels weighing 48 pounds each ; and 2 sacks 
of oats containing 2 bushels each, and each bushel weighing 
32 pounds ? 

12. In one box there were 8 layers of 32 eggs each ; in 
another box, 10 layers of 32 eggs each. How many eggs 
were there in the 2 boxes ? 

13. Find the cost of the following : 2 horses, $125 each ; 
2 wagons, $75 each ; and harness for 2 horses, $24 each. 

ljj,. How much gas was used in a house in 4 months, where 
the numbers of cubic feet of gas used were as follows : 2,900 ; 
3,800; 4,600; and 2,500 ? 

15. Mr. Shaw paid for five bills of goods the following 
amounts: $9,950; $7,648; $8,209; $10,678; and $7,896. 
What was the entire amount paid ? 

16. Mr. Harrison in one month drew 8 checks on his 
bank for the following sums : $125, $368, $724, $432, $602, 
$900, $783, and $98. What was the entire amount ? 

17. A builder paid wages to laborers as follows : $13 to 
each of 9 men ; $22 to each of 8 men ; $18 to each of 3 men ; 
and $23 to each of 5 men. How much in all did he pay ? 

18. A merchant sold in one year 1986 barrels of apples ; 
2978 barrels of potatoes; 3369 barrels of flour; and 678 
barrels of sugar. How many barrels altogether did he sell ? 



110 SUBTBACTIOJSf. 

SUBTRACTION. 

TWENTY TO FORTY. 

223. 1. Frank had to travel a distance of 31 miles. How 
many miles remained after he had traveled 1 mile ? 2 miles ? 
3? 4? 

2. Martha's Second Header cost 31 cents. How much 
money did she have left, if the amount was 5 cents less than 
the cost of the reader ? 6 cents less ? 7 cents ? 

3. Benjamin had 3 dimes and 1 cent ; Fred had 8 cents 
less than Benjamin ; Emma, 9 cents less ; and John, 10 cents 
less. How much money did Fred, Emma, and John each 
have ? 

Jf. How much money will remain in a purse containing four 
$5 bills and six $2 bills after $5 are spent ? After $6 are 
spent, how much will remain ? $7 ? 

5. A farmer had 32 quarts of berries. How many did he 
have after selling 8 quarts ? 9 quarts ? 10 quarts ? 

6. John now earns $33 per week. How much did he 
earn when he earned this amount less $3 ? $4 ? $5 ? $6 ? 

7. There are 33 pupils in a class. How many will remain, 
if during the year the number discharged is 7? 8? 9? 10? 

8. Mr. Higgins was 35 years old. His wife was 7 years 
younger. How old was his wife ? 

9. John started to walk 36 miles. After walking 7 miles, 
how many miles did he still have to walk ? 

10. Mr. Jackson earns $20 one day and $16 the next. He 
then pays $5 for his hotel bill and $3 for a hat. How much 
has he left ? 

11. Mamie had 37 cents. She spent 7 cents for candy and 
5 cents for an orange. How much did she then have ? 

12. Frank had 38 miles to travel. After skating 9 miles 
of the distance, how much farther must he travel to complete 
his journey ? 



TWENTY TO FORTY. Ill 

224. Announce differences at sight: 

Read the answers : 1. From left to right; 2. From right to left; 3. 
From top to bottom; 4. From bottom to top; 5. As directed by the 
teacher. 

1st. 2d. 3d, 4th. 5th. 6th. 7th. 8th. 9th. 

(31 31 31 31 31 31 31 31 31 

A jl 3 5 8 10 7 64 2 



B 



C 



D 



E 



34 


34 


34 


34 


34 


34 


34 


35 


35 


4 


6 


9 


10 


8 


7 


5 


5 


6 


32 


33 


33 


33 


33 


33 


33 


33 


33 


7 


3 


10 


4 


9 


5 


8 


6 


7 


35 


35 


35 


35 


36 


36 


36 


36 


36 


7 


9 


8 


10 


6 


9 


7 


8 


10 


31 


32 


32 


32 


32 


32 


32 


32 


32 


9 


2 


4 


6 


9 


8 


10 


5 


3 


37 


37 


37 


37 


38 


38 


38 


39 


39 


7 


10 


9 


8 


8 


10 


9 


9 


10 



225. Write from dictation and find the differences ; 



1. 


». 


3: 


4. 


s. 


8,428 


7,504 


5,400 


7,501 


3,207 


6,315 


3,283 


2,700 


3,438 


1,838 


6. 


7. 


8. 


9. 


lO. 


6,431 


5,050 


9,000 


9,090 


9,400 


3,576 


1,409 


8,899 


5,507 


9,299 



112 SUBTRACTION. 



WRITTEN PROBLEMS. 



226. i. New York was settled in the year 1614 ; Ohio, in 
1788. How many years before Ohio was New York settled ? 

2. The Kio Grande is 1,800 miles long ; the Hudson River, 
325 miles long. How much longer is the Rio Grande than 
the Hudson ? 

3. Connecticut has an area of 4,990 square miles ; Ver- 
mont, 9,565 square miles. Which is the larger, and how 
much ? 

Jf. The city of Portsmouth contained 9,732 people, when 
Carson City contained 4,850 people. What was the difference 
between the populations of the two cities ? 

5. Which is the higher and how much, Mount Marcy in 
New York, 5,379 feet high, or Mount Mansfield in Vermont, 
4,430 feet high ? 

6. How many years is it since 
the Mayflower landed at Plym- 
outh Rock in the year 1620 ? 

7. Mr. Howe's farm is worth 
$8,300. Mr. White's farm is 
worth $6,580. What is the differ- 
ence between the values of their 
farms ? 

8. A gentleman having $6390 

in bank drew a check for $3475. What balance did he then 
have in bank ? 

9. A stock of dry goods was bought for $9,420 ; it was sold 
for $10,030. How much was gained by the sale ? 

10. How many years is it since the Constitution of the 
United States was adopted in 1789 ? 

11. If 1,049 acres of land are sold from a farm containing 
5,750 acres, how many acres will remain ? 

12. Mr. Harmon bought a house for $17,500; he paid 
$8,750. How much of the purchase price remained unpaid ? 




TWENTY TO FORTY. 



113 



4 x 6 = 24 
6 x 4 = 24 


24 +- 4 = 6 
24 ^ 6 = 4 


4x7=28 
7 x 4 = 28 


28 h- 4 = 7 
28 ^ 7 = 4 


4x8 = 32 
8x4 = 32 


32 -^ 4 = 8 
32 ^- 8 = 4 


4 x 9 = 36 

9x4 = 36 


36 ^ 4 = 9 

36-f-9 = 4 



MULTIPLICATION AND DIVISION. 

TWENTY TO FORTY. 

237. i. When the fare 
for one passenger is 6 cents, 
how much is the fare for 4 
passengers ? 

2. If a street-car ticket 
costs 4 cents, what will 6 
cost ? 

3. If 24 cents will buy 
4 packages of envelopes, 
how much will buy one 
package ? 

Jf. A young man paid $24 
for 6 weeks' board. How much did he pay per week ? 

5. In one week there are 7 days. How many days are 
there in 4 weeks ? 

6. If one pint of milk costs 4 cents, what will 7 pints cost ? 

7. 28 yards of calico will make how many dresses contain- 
ing 7 yards each ? 

8. How many gallons of milk are there in a can contain- 
ing 28 quarts ? 

9. If a quarter of a yard of lace costs 8 cents, what does 
a yard cost ? 

10. At 4 cents a pint, what will 4 quarts of cider vinegar 
cost ? 

li. For 32 cents, how many oranges can you buy at 4 
cents each ? At 8 cents each ? 

12. If each of 4 bunches contains 9 grapes, how many 
grapes are there in the 4 bunches ? 

13. What will be the cost of 9 gallons of alcohol at $1 a 
quart ? 

1J±. $36 were received as pay for 4 weeks' work. How 
much was paid per week ? 



6. 


7. 


s. 


4 


9 


4 


8 


4 


9 



114 MULTIPLICATION AND DIVISION 

228. State products at sight : 

i. 2. 3. 4. 5. 

6 4 7 4 8 

_4 _ 6 _i _J ± 

229. State quotients at sight: 

1. 2. 3. 4. 5. 6. 7. 8. 

4)24 6)24 4)28 7)28 4)32 8)32 4)36 9)36 

230. Copy and multiply: 



Explanation. — 1. 396 is first multi- 
plied by the 4 units, giving for a prod- 
uct 1584 units, which is written so that 
G) / the right-hand or units' figure stands in 

units' place. 



3Cj6 



7Cj2 



/ ^)& / ®' 396 * s ^ en mu l ti P^ e( l by 2 tens, 

^ giving for a product 792 tens, which is 

written so that the right-hand figure 
stands in tens' place. 
/i p? /~i i a 3. Adding the two products together 

/ ) *-* " ^ • gives the entire product or answer, 9, 504. 

2. 3. 4. 5. 6. 

478 563 798 627 358 
32 34 24 43 44 



7. 8. 9. 10. 11. 

679 243 324 413 332 

34 38 , _89 _37 _98 

231. Write from dictation and divide : 

l. 2. 3. 4. f>. 

4)736 6)258 8)352 9)297 3)972 

6. 7. 8. 9. 10. 

7)294 3)894 8)264 7)301 9)378 



TWENTY TO FORTY. 115 



WRITTEN PROBLEMS. 

232. 1. If 4 equal loads of flour cost $288, what is the 
cost of one of the loads ? 

2. In each of 24 cargoes of coal there were 275 tons. How 
many tons altogether were there ? 

3. If coal for one winter is bought for $164, how much 
should be paid for coal enough to last 4 winters ? 

4* If a farm containing 938 acres is divided equally among 
7 families, how many acres does each family receive ? 

5. If each of 234 acres in a field yields 73 bushels of corn, 
how many bushels altogether will there be ? 

6. If a farm containing 124 acres of land is sold for $64 
per acre, for how much is the farm sold ? 

7. If a factory burns 9 tons of coal in a day, how many 
tons will it burn in 313 days ? 

8. A steamer used 287 tons of coal in 7 days. How many 
tons did it use per day ? 

9. A city lot containing 34 front feet was sold for $175 per 
foot. What was the selling price of the lot ? 

10. How long will the water in a river be in flowing 168 
miles, if it flows 7 miles per hour ? 

11. At $6 each, how many silk hats can be purchased for 
$252 ? 

12. How much will 
9 sheep weigh, if 
each sheep weighs 123 
pounds ? 

13. How far will a 
steamship sail in 245 

hours, if it sails 24 miles each hour ? If it sails 25 miles ? 

14- If a man earns $8 per day, how long will it take him 
to earn $336 ? 

15. What will be the cost of 29 city lots, at $1432 each ? 

10. If 8 tons of hay cost $112/ what (Joes each ton cost ? 




116 



MULTIPLICATION AND DIVISION 



5 x 5 = 25 


25 -h5 = 5 


5x6 = 30 
6x5 = 30 


30 -s- 5 = 6 
30 -f- 6 = 5 


5x7=35 
7x5 = 35 


35 -*- 5 = 7 
35 -s- 7 = 5 


6 x 6 = 36 


36 -J- 6 = 6 



TWENTY TC FORTY. 

233. 1. At 5 cents each, 
what will 5 rides in a street 
car cost ? 

^. How many oranges at 
5 cents each can be bought 
for 25 cents ? 

3. What will be the cost 
of 5 trees at $6 each ? 

Jf. What would be the 
cost of 6 trees at $5 each ? 

5. If there are 6 working days in a week, how many weeks 
would give 30 working days ? How many weeks would give 
24 working days ? 

6. If it costs $5 a day to live in a hotel, for how many 
days will $30 pay ? 

7. In one week there are 7 days. How many days are 
there in 5 weeks ? 

8. At $5 a day, how much would it cost to live at the sea- 
side for one week ? 

9. For 35 cents, how far can you travel by railroad at 5 
cents per mile ? 

10. If one man can do a piece of work in 35 days, how 
long will it take 7 men to do the work ? 

11. Every lily has 6 petals. How many petals have 6 
lilies ? 

12. In a class there are 36 pupils arranged in 6 equal rows. 
How many pupils are there in each row ? 

13. If ^ yard of lace costs 6 cents, w T hat will half a yard 
cost ? A yard and a half ? 

ljf. If cheese sells for 16 cents a pound, what will a quar- 
ter of a pound cost ? A pound and a quarter ? 

15. At 9 cents a dozen, what is the cost of a third of a 
dozen buttons ? One dozen and a third ? 



TWENTY TO FORTY. 117 

16. At 6 cents a yard, what will six and a half yards of 
ribbon cost ? 

Six and a half times 6 cents equals 39 cents. 

6* x 6i = sp. 

17. What will be the cost of four and a third dozen 
buttons at 9 cents a dozen ? 

Four and a third times 9 cents equals 89 cents. 

tjt x 4- 3 = ^ 

i£. At 8 cents a pound, what will a quarter of a pound of 
sugar cost ? One pound and a quarter ? 

19. What will be the cost of four and a quarter pounds 
of crackers at 8 cents a pound ? 

Four and a quarter times 8 cents equals 3Jf cents. 

8t x 4i = 34* 



234. State products at sight : 








1. S. 3. 4. 


5. 


6. 


7. 


6 4 8 4 


7 


5 


7 


4 6 4 8 


5 


7 


4 



s. 


9. 


lO. 


n. 


12. 


is. 


14 


4 


5 


9 


4 


6 


5 


6 


_7 


5 


4 


9 


5 


6 


6 



235. State quotients at sight: 

1. 2. 3. 4. 5. 6. 7. 

4)24 6)24 4)32 8)32 5)30 6)30 5)25 

S. 9. 10. II. 12. 23. J4. 

4)28 7)28 4)36 9)36 5)35 7)35 6)36 



118 MULTIPLICATION AND DIVISION 

236. Find the value of: 

1. 6x6 — 8 + 3 — 9 — 6 + 5. s. 9^3 — 2 + 9 — 7 + 9 + 6. 

2. 5x5 — 7 + 4—7 + 9 + 7. 6\ 25-^5 + 8 + 7 — 6 + 9 — 7. 
s. 4x4 + 6 + 9 + 4 — 8 — 9. 7. 16-^-4 + 9 + 8 — 5 — 7 + 8. 
4. 3x3 + 7 + 8 — 9 + 8-7. *. 36-^6 + 9 + 8 + 9-7 — 8. 

237. Copy and multiply: 

*• Explanation.—/. One half of 286 

G) Q A units is 143 units, which is written so that 
the right-hand or units' figure stands in 

$? J? - units' place. 

2 2. 286 multiplied by 3 units gives 858 

/ / units, which is written so that the right- 

^ hand or units' figure stands in units' 

858 p lace - 

3. 286 multiplied by 2 tens gives 572 
O / sd tens, which is so written that the right- 

7 hand or tens' figure stands in tens' place. 

U 724 AnS. 4' Tne different products added to- 

gether give for the answer, 6721. 



3. 


3. 


4. 


5. 


6. 


369 


484 


862 


264 


963 


32i 


_24| 


m 


56i 


33^ 


7. 


s. 


9. 


10. 


11. 


242 


844 


693 


872 


396 


68J 


44i 


Jit 


25| 


42| 



238. Write from dictation and divide : 

l. 2. 3. 4. 5. 

5)755 6)372 7 )294 8 )336 9 )207 

6. 7. 8. 9. 10 

9)378 7)308 5)380 8)256 6)270 



TWENTY TO FORTY. 119 

WRITTEN PROBLEMS. 

239. 1. What will be the cost of 43^ acres of land at 
$682 per acre ? 

2. If there are 43^ pounds of candles in one box, how many 
pounds will there be in 369 boxes ? 

3. If 352 men march in 8 equal lines, how many men are 
there in each line ? 

J/,. A man owing a bill of $575 paid one fifth of it. How 
much did he pay ? 

5. In one barrel of oil there are 31^ gallons. How many 
gallons are there in 684 barrels ? 

6. There are 16| feet in one rod. How many feet are 
there in 284 rods ? 

7. In 9 months a man saved $1278. How much did he 
save per month ? 

8. If in each piece of cloth there are 43^ yards, how many 
yards are there in 639 pieces ? 

9. What will be the cost of 248. suits of clothes at $24J a 
suit ? 

10. If a man has 364 miles to walk in 7 days, how many 
miles must he walk each day ? 

11. If 8 partners make equal parts of $3472, how much 
does each partner make ? 

12. In 9 months a boy earned $396. How much was this 
per month ? 

13. Find how many passengers a train of 18 excursion 
cars will seat, if each car has seats for 124. 

1J±. What is the entire weight of 125 sacks of wheat, 
each containing 2 bushels, and each bushel weighing 60 
pounds ? 

15. If there are 25 bushels in a load of coal, and each 
bushel weighs 80 pounds, what will 2 loads weigh ? 

16. How much does each excursion ticket to California 
cost, if 4 of them cost $304 ? 



120 



MEASUREMENT. 



MEASUREMENT. 

GRAINS, SEEDS, ETC. 




2 pints 


make 


1 quart. 


8 quarts 


(( 


1 peck. 


4 pecks 


a 


1 bushel. 



ORAL PROBLEMS. 

240. i. How many pints Dry Measures, 

are there in a peck ? 

2. How many quarts are 
there in a bushel ? 

3. If a pint of chestnuts is 
worth 5 cents, what is a quart 
worth ? 

Jf. If potatoes cost 20 cents a peck, what is the price of a 
small measure or half a peck ? 

5. If a quart of berries costs 8 cents, what will a peck cost ? 

6. When potatoes sell for one dollar a bushel, how much 
will a peck of- potatoes cost ? 

7. If half a peck of apples is worth 10 cents, what is the 
value of 2 pecks of apples ? 



WRITTEN PROBLEMS. 

241. 1. Find how many pints there are in a bushel. 
£. If raspberries sell for 4 cents a pint, at the same rate 
what will a bushel cost ? 

3. How many bushels are there in 976 pecks ? 



MEASUREMENT. 



121 



WEIGHTS. 




Avoirdupois Weight. 



16 


ounces 


make 


1 pound. 


lOO 


pounds 


a 


1 hundredweight. 


20 


hundredweights 


a 


1 ton. 



ORAL PROBLEMS. 

242. i. How many ounces are there in half of a pound ? 

2, How many ounces are there in a quarter of a pound ? 

3. If a pound is divided into eight equal parts, how many 
ounces will there be in each part ? 

Jf. If 4 ounces of candy cost 6 cents, what will a pound 
cost ? 

5. If a pound of sugar is worth 8 cents, what should 8 
ounces cost ? 

6. If a pound of raisins is worth 16 cents, what are 4 
ounces worth ? 

7. How many hundredweight are there in half a ton ? 



WRITTEN PROBLEMS. 

243. i. In 462 pounds how many ounces are there ? 

2. In 4 tons how many hundredweights are there ? 

3, At 1 cent an ounce what are 362 pounds of steel worth ? 
4- If an ounce of needles is worth $1, what are 254 pounds 

worth ? 

5. If a ton of hay is worth $18, what are 432 J- tons worth ? 

6. A pound of sirloin steak is worth 22 cents. What are 
8 ounces worth ? 



SECTION IV. 



DEFINITIONS. 

244. A unit is a single thing or one ; as one apple, one 
dollar, one hour, one. 

A group of things, if considered as a single thing or one, is also a 
unit ; as one class, one dozen, one ten. 

245. A number is a unit or a collection of units ; as seven 
apples, five dollars, six classes, eight. 

246. The unit of a number is one of the collection. Thus, 
one apple is the unit of seven apples ; one dollar of five dol- 
lars ; one of eight. 

2>4tH. A concrete or denominate number is a number 
whose units are named ; as five pounds, seven books. 

248. An abstract number is a number whose units are 
not named ; as three, six, eleven. 

A concrete or denominate number expresses some particular kind of 
quantity. An abstract number expresses one or more units without 
reference to any particular object, or objects. 

249. Like numbers are numbers whose units are the 
same ; as, $6 and $9 ; 7 cts. and 8 cts. ; or, 8 men and 9 men. 

250. Unlike numbers are numbers whose units are dif- 
ferent ; as, $6 and 9 men ; 9 cts. and 8 tops ; or, 5 apples 
and 6 oranges. 

251. Arithmetic is the science, which treats of numbers 
and the art, which treats of their applications. 

As a science, arithmetic treats of the properties and relations of num- 
bers ; as an art, it treats of their use. 



NOTATION AND NUMERATION. 123 



NOTATION AND NUMERATION. 

252. Numbers are expressed in three ways : 1. By ivords ; 
2. By figures ; 3. By letters. 

253. Notation is the art of expressing numbers by 
figures, or letters. 

254. Numeration is the art of reading numbers ex- 
pressed by figures, or letters. 

ARABIC SYSTEM. 

255. In the Arabic System of Notation, ten figures are 
employed to express numbers, viz. : 

123456789 0. 

One, Two, TJiree, Four, j&ive, Six, Seven, Eight, Nine, Naught. 

256. The first nine of these figures are called significant 
figures or digits, and each expresses the number of units 
indicated by its name. 

257. The last figure, naught, also called cipher or zero, 

stands for nothing. It is used with the other figures in 
expressing numbers larger than 9. 

Figures are not numbers but are characters used to express 
numbers. By using them in accordance with certain principles, any 
number may be expressed. 

258. Nine (9) is the largest number that can be expressed 
by a single figure. To express ten units, or 1 ten, the figure 
1 is written at the left of the cipher ; thus, 10. 

259. Ninety-nine (99) is the largest number that can be 
expressed by two figures. To express one hundred units or 
10 tens, the figure 1 is written in the third place, or at the 
left of two ciphers ; thus, 100. 



124 NOTATION AND NUMERATION 

260. Nine hundred ninety-nine (999) is the largest num- 
ber that can be expressed by three figures. To express one 
thousand units, 10 hundreds, the figure 1 is written in 
the fourth place, or at the left of three ciphers ; thus, 1000. 

261. Numbers of more than three figures are divided into 
periods of three figures each, commencing at the right hand. 

TABLE OF NOTATION AND NUMERATION. 

Names of 4th. 3d. 2d. 1st. 

Periods. Billions. Millions. TJiousands. Units. 



Names of ., 
Places. 



12th.llth.Wth. 9th. 8th. 7th. 6th. 5th. 4th. 3d. 2d. 1st. 






S 9 *3 fl 



^CD— ^3^ 2 <£> fl , R O ^ 

W H U> Wh^ W h P ffl h £ 
Number. 180,70 0,520,234 



262. 1. Each period is read as if it formed a number by 
itself, and then the name of the period is read. 

2. The above number is read one hundred eighty billion, 
seven hundred million, five hundred twenty thousand, two 
hundred thirty-four. 

Note. — Observe that in reading any given whole number, the final 
letter, s, is omitted in naming the period ; units' period is not named ; 
and the word and is not used. 

263. Rule for Notation. — Begin at the left-hand 
and write the hundreds, tens, and units of each pe7*iod 
in succession, filling vacant orders and periods with 
ciphers. 

264. Rule for Numeration. — I. Separate the num- 
ber into periods, beginning at the right. 

II. Begin at the left and read each period as if it 
stood alone, adding its name. 



NOTATION AND NUMERATION 125 

265. Read the following numbers: 

i. 4,239. 6. 340,765. n. 16,054,320. 

s. 6,048. 7. 550,052. ■ 12. 36,290,011. 

3. 9,091. 8. 676,760. is. 26,000,560. 

4. 12,006. 9. 1,100,110. 14. 12,012,120. 

5. 20,201. 10. 2,000,200. 15. 25,250,025. 

266. Empress the following numbers in figures: 

1. Two thousand, seventy-six. 

2. Four thousand, three hundred eighty. 

3. Eleven thousand, one hundred ten. 
Jf. Twenty thousand, twenty-one. 

5. Forty-nine thousand, four hundred ninety. 

6. Fifty thousand, five hundred. 

7. One hundred thousand, one hundred. 

8. Three hundred thousand, three. 

9. Four hundred one thousand, forty. 

10. Five hundred fifty thousand. 

11. Six hundred thousand, sixty. 

12. Seven hundred seven thousand, seven. 

13. Eight hundred thousand, eighty-one. 
ljj,. One million, ten thousand, one hundred. 

15. Ten million, one hundred ten thousand. 

16. Fifty million, five hundred five. 

267. Write in words the numbers in Article 265. 

268. Write from dictation the numbers in Articles 
265 and 266. 

269. The sign of dollars is $ ; and cents are denoted 
by ct., $, cts., or by placing a period between dollars and 
cents. 

Thus, $24 and 1 ct. ; $24 and lit, or $24 and 17 cts. ; or $24.17. 
There are 100 cents in one dollar; hence, cents must always occupy 
two places. Thus, $5 and 8 cts. are written $5.08. 



126 NOTATION AND NUMERATION 

270. Head the following sums of money : 

1. $12.25. 5. $20.02. 9. $1050.50. 

2. $103.32. 6. $203.30. 10. $2002.02. 

3. $500.56. 7. $440.04. n. $7070.70. 

4. $700.07. s. $600.60. 12. $10010.10. 

271. Write the following sums of money: 

1. Five dollars and fifty-five cents. 

2. Six dollars and six cents. 

3. Seventy dollars and seventy cents. 

Jf. Eighty hundred eighty-six dollars and eighty-six cents. 

5. Nine hundred dollars and ninety cents. 

6. Nine hundred dollars and nine cents. 

7. One thousand ten dollars and eight cents. 

8. Ten thousand dollars and four cents. 



ROMAN NOTATION. 

272. In the Roman System of Notation, seven let- 
ters are employed to represent numbers, viz. : 



Letters. 


I. 


V. 


X. 


L. 


0. 


D. 


M. 


Values. 


1 


5 


10 


50 


IOO 


500 


IOOO 



273. In combination, their significance is in accordance 
with the following 

PRINCIPLES. 
I. To repeat a letter repeats its value. 
Tims, III. is 3; XXX. is 30; CCCC. is 400; MMMM. is 4000. 

The letters Y, h f and D are never repeated. 

The letter I or X may be used three times in any one place. 

The letter C or M may be used four times in any one place. 



NOTATION AND NUMERATION 



127 



II. When, one letter follows another of greater value, 
the sum of their values is denoted ; when it precedes, 
the difference between their values is denoted. 

Thus, XXV. is 25; XL. is 40; XC. is 90; CX. is 110. 

III. A letter between two of greater value diminishes 
by its value the sum of the values of the other two 
letters. 

Thus, XIX. is 19; LXIV. is 64; DXC. is 590. 

IV. A bar over a letter increases its value one thou- 
sand times. 

Thus, V. is 5000, and M. is 1000000. 

Note. — Roman numbers are always followed by a period. 

274. Express by figures, or the Arabic Notation: 



i. XIX. 

2. XVIII. 

3. XIV. 

4. XXI. 

5. XXIV. 



s. XL. 

». XLIV. 
io. XLVIII. 
n. XLIX. 
is. LXIX. 



is. XCI. 
w. XCIX. 
it. OIL 
is. CXLIX. 
i'j. CCIV. 
so. CCCXC. 



22. CCCC. 

23. DCL. 

24. DCCX. 

25. MDL 

26. MCCCC. 

27. VDLV. 



e. XXVI. is. LXXIV. 

7. XXIX. a. LXXXIII. 2i. C00X0IX. 2s. MDCCC. 



275. Express by letters, or the Roman Notation : 



i. 17. 


8. 


75. 


is. 201. 


22. 


888. 


2. 27. 


9. 


89. 


16. 240. 


23. 


1006. 


3. 31. 


IO. 


91. 


17. 303. 


24. 


1066. 


4. 36. 


11. 


94. 


is. 348. 


25. 


1666. 


5. 41. 


12. 


101. 


io. 405. 


26. 


6666. 


6. 49. 


13. 


119. 


20. 455. 


27. 


1887. 


7. 55. 


14. 


149. 


2i. 509. 


28. 


100000. 



128 ADDITION, 

ADDITION. 

1. If Thomas earns $9 and his brother Edgar $7 per week, 
how many dollars do both earn ? $9 and $7 are how many 
dollars ? 9 and 7 are how many ? 

2. How many books has Fred on two shelves, if he has 17 
books on one shelf and 8 on the other ? 

276. The sum of two or more numbers is a number con- 
taining as many units as the numbers taken together. 

277. Addition is the process of finding the sum of two 
or more numbers. 

2. Can you add 8 cts. and 7 cts. ? What kind of numbers 
are they ? Can you add $5 and 5 lb. ? What kind of num- 
bers are they ? 

278. Principles. — I. Only like numbers, and units of 
the same order, can be added. 

II. The sum is like the numbers added. 

279. To find the sum of ttvo or more numbers ex- 
pressed by any number of figures. 

1. Find the sum of 4,049; 3,787; 2,089; 1,847; 3,668. 

Process. Explanation. — 1. The numbers 

/ /) / /9 are written so that units of the same 

+r-) " -H^ / order stand in the same column. 

3 78 7 2 ' The sum of the units is 40 ' equal 

'■ to 4 tens and units ; the is written 

^ (J O U under the units' column, and the 4 

/ O / >y ^ ens are carr i e d to the tens' column. 

J yO X^f s. The sum of the tens, including 

*? U N ^s those from the units' column, is 34, 

^ equal to 3 hundreds and 4 tens; the 



A F\ / / fl a ^ is written under the tens' column, 

) -tr-^t ' an( i the 3 hundreds are carried to the 

hundreds' column. 
4. In the same manner the sum of each successive column is found. 



ADDITION. 129 

280. Rule. — I. Write the numbers so that units of 
the same order shall stand in the same eolumn. 

II. Begin at the right and add each column sepa- 
rately. When the sum is less than ten, write it under 
the column added ; when the sum is ten or more than 
ten, write the units' figure under the column added 
and carry the ten or tens to the next column. 

Proof. — Find the sum by adding the columns in the 
opposite direction ; if the results agree the work may be 
considered correct. 

281. Write from dictation and find the sums : 



1. 


2. 


3. 


i. 


5. 


2,878 


1,909 


2,541 


1,656 


5,541 


3,509 


2,786 


3,698 


2,487 


2,028 


1,687 


1,598 


2,785 


3,759 


3,170 


2,196 


2,179 


3,989 


2,887 


1,099 


3,857 


3,397 


1,657 


1,789 


1,900 


6. 


7. 


s. 


9. 


10. 


2,398 


1,914 


5,009 


2,207 


1,091 


4,763 


1,008 


2,980 


3,086 


2,347 


1,589 


3,879 


3,075 


1,973 


5,726 


2,647 


2,058 


4,168 


2,058 


2>431 


1,598 


1,700 


9,088 


1,602 


3,168 


n. 


12. 


13. 


14. 


is. 


9,932 


1,197 


4,053 


5,500 


4,411 


2,397 


7,754 


1,808 


2,020 


20,640 


3,462 


3,168 


2,660 


13,760 


3,781 


1,588 


8,659 


3,541 


9,289 


2,159 


2,673 


2,730 


12,276 


2,175 


10,011 



282. The following exercise, which requires pupils to 
give the sum of two numbers at sight, is designed to secure 
rapidity and accuracy in the addition of long columns, 



130 ADDITION. 

283. Announce sums at sight : 

Read answers : 1. From left to right ; 2. From right to left ; 3. From 
top to bottom ; 4. From bottom to top ; 5. As directed by the teacher. 

1st. 2d. 3d. Jfth. 5th. 6th. 7th. 8th. 9th. 
(832976423 



B 



C 



D 



E 



F 



G 



H 



u 


4 


3 


2 


8 


9 


9 


8 


5 


\i 


5 

2 


6 
5 


8 
4 


7 
2 


1 
1 


1 
6 


4 
5 


5 

7 


\i 


1 
3 


5 

1 


8 
3 


7 
4 


6 
6 


5 
4 


4 

7 


5 

8 


\l 


1 

8 


3 
3 


6 

2 


4 

4 


6 

7 


9 

8 


7 
9 


1 

2 



6 5 2 3 5 4 231 

_3_9_4_8J3_1_2_6_9 

3587656 9 8 

236785468 



9 


2 


4 


1 


3 


8 


9 


4 


2 


3 


1 


2 


5 


7 


9 


4 


6 


9 


1 


9 


8 


9 


8 


9 


7 


1 


6 


7 


9 


7 


5 


2 


7 


6 


4 


2 


6 


2 


9 


7 


3 


2 


8 


7 


7 


1 


7 


1 


5 


1 


5 


1 


3 


1 



Note.— The attainment of rapidity should be regarded as of primary 
importance. 



ADDITION. 131 

ORAL PROBLEMS. 

384, 1. John has 33 cts. and finds 8 cts. more. How 
many has he then ? 3 + 8=? 33 + 8=? 

2. Mary spends 39 cts. for lace and 8 cts. for thread. How 
much does she spend ? 9 + 8=? 39 + 8=? 

3. Jane picked 38 quarts of berries, and Kate 5 quarts. 
How many quarts did both pick ? 8 + 5=? 38 + 5=? 

4* Henry rode 39 miles in the cars, and walked 9 miles. 
How far did he travel ? 9 + 9=? 39 + 9=? 

5. How many apples are 34 apples, 9 apples, 7 apples, and 
8apples? 4 + 9=? 34 + 9=? 

6. A gentleman pays board as follows : for himself and 
wife $38 per week, and for the servant $6 per week. What 
is his bill by the week ? 8 + 6=? 38 + 6=? 

7. Philip spent $33 for a suit of clothes, $12 for a pair of 
boots, and $9 for a cane. How much did all cost ? 

8. Emma spent 45 cents for muslin, 15 cents for lace trim- 
ming, and 7 cents for elastic. How much money did she 
spend ? 

9. Mr. Brooks sold 4 hats at $6 each, and 3 umbrellas at 
$3 each. How much money did he receive in all ? 

10. If Edgar walks one day 4 miles an hour for 9 hours, 
and the next day 5 miles an hour for 2 hours, how far will he 
walk in both days ? 

11. If a ticket from New York to San Francisco sells for $79, 
ticket from San Francisco to Portland 

lor $12, how much will both tickets cost? 

12. If each of a party of 3 Indians 

shoots 4 deer, and each of a 

party of 2 Indians shoots 2 

deer, how 

many deer 

in all will 

they shoot? 




132 ADDITION. 

285. Announce sums at sight: 

Read answers: 1. From left to right; 2. From right to left; 3. From 
top to bottom ; 4. From bottom to top ; 5. In any order suggested by the 
teacher. 

1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 

tyiyrytyiyiyiyiyiy 

"UO 30 60 90 20 40 70 80 50 



^41 21 31 11 91 71 61 51 81 



tyiytytyiytyiytyty 

"^22 12 52 42 32 62 92 82 72 



wryiyiyiy'tviytyiy 

"^93 63 33 13 23 53 83 43 73 



777777777 
^84 94 54 44 34 24 14 64 74 



777777777 
^65 45 25 35 15 55 95 85 75 



G 



^777777777 
^16 4676962636566686 

777777777 
^27 57 87 17 37 67 47 77 97 



7 7 7 7.7 7 77 7 

"*58 48 68 38 ' 78 28 88 18 98 



777777777 
M9 49 99 29 89 39 79 59 69 



ADDITION. 133 

286. The preceding is a model for a series of exercises to 
supplement the one on page 130. 

287. Each of the other exercises in the series is made by 
substituting for 7, one of the other digits. When prepared, 
each exercise should be faithfully practiced. The utility of 
these exercises appears when the operation of adding a long 
column is considered. 

288. Find the sums of the following : 

1st. 2d. 3d. Jfth. 5th. 6th. 7th. 8th. 9th. 10th. 
25 3 1942654 



5 


3 


1 


6 


1 


3 


7 


3 


9 


8 


3 


2 


2 


7 


8 


5 


6 


2 


7 


9 


4 


6 


4 


6 


2 


2 


3 


7 


6 


7 


6 


4 


3 


5 


3 


6 


5 


4 


8 


6 


1 


2 


1 


4 


7 


1 


2 


8 


2 


8 


7 


3 


3 


3 


2 


9 


6 


1 


3 


7 





2 


6 


2 


3 


8 


4 


5 


7 


9 


3 


1 


5 


1 


5 


2 


3 


2 


4 


5 


2 


7 


2 


3 


4 


7 


5 


3 


8 


3 


3 


3 


1 


5 


1 


3 


1 


4 


7 


8 



289. In practice, all that should be said or thought at 
each step, in the addition of a column of figures, is simply 
the amount. Thus, 

The first of the above columns is added by thinking or saying 8, 13, 
16, 18, 21, 23, 26, 33, 34, 44, 47, 52, 54; and one proficient in adding will 
add quite as fast as he can read. 



134 



4 




ADDITION. 




290. Write from 


dictation and add the following : 


l. 


2. 


3. 


4. 


5. 


$4.35 


$938 


$359 


$8.23 


$5.39 


2.68 


872 


547 


3.67 


1.92 


1.95 


594 


926 


5.48 


3.84 


3.48 


638 


185 


3.29 


4.79 


8.65 


397 


378 


1.76 


2.88 


6.96 


846 


299 


2.83 


6.54 


2.33 


378 


188 


3.57 


1.98 


1.89 


199 


377 


2.88 


2.37 


$ 


$ 


$ 


$ 


$ 


6. 


7. 


8. 


9. 


10. 


18 


2,002 


$456.32 


$202.20 


$1,100.50 


755 


189 


327.48 


110.01 


238.09 


49 


1,047 


109.63 


37.07 


1,304.40 


3,684 


763 


200.58 


6.90 


886.53 


873 


3,030 


99.37 


1,480.30 


2,775.48 


199 


847 


100.11 


375.83 


653.00 


237 


38 


48.33 


456.51 


77.70 


7,408 


4,909 


303.03 


273.38 


1,540.04 






ORAL PROBLEMS. 





291. 1. Tillie bought fish for 38 cents and a bag of salt 
for 5 cents. What did both cost ? 

2. Fred traveled in four days the following distance : 
38 miles, 9 miles, 8 miles, and 6 miles. How far did he 
travel ? 

3. If Jacob catches 48 trout, Henry 10, Charles 8, and 
Edward 6, how many trout do they all catch ? 

Jf. If an arithmetic costs 53 cents, a speller 20 cents, and a 
slate 8 cents, what is their entire cost ? 

Note. — The pupils should be required to construct oral problems, and 
to write them neatly in correct language. Problems so made may fre- 
quently be used with profit for a class exercise. 




ADDITION. 135 



WRITTEN PROBLEMS. 

292, i. What is the united weight 
of 4 whales weighing respectively 50,900, 
48,800, 59,700, and 60,800 pounds ? 
2. The brown-stone fronts to four different 
nouses cost: $1,750; $1,438; $1,659; $1,537. 
What did the four fronts together cost ? 

3. In five different months a lady purchased the following 
bills of goods at a dry-goods store : $84.37 ; $72.35 ; $98.64 ; 
$87.53 ; $106.60. How much did she purchase ? 

Jf.. Find the cost of the following building material : 1 
barrel of plaster, $2.05 ; 1 barrel of finishing lime, $2.00; 1 
barrel of common lime, $1.30 ; 1 load of sand, $1.00. 

5. Find the entire cost of the following : 10,000 Bal- 
timore bricks, $420; 20,000 Philadelphia bricks, $800; 
5,000 Pittsburgh bricks, $145 ; and 100,000 common bricks, 
$650. 

6. A butcher sold 4,860 pounds of beef, 3,768 pounds of 
pork, 2,568 pounds of mutton, and 886 pounds of veal. How 
many pounds of meat did he sell ? 

7. Maine contains 33,040 square miles ; New Hampshire, 
9,305 square miles; Vermont, 9,565 square miles; Massa- 
chusetts, 8,315 square miles ; Rhode Island, 1,250 square 
miles; and Connecticut, 4,990 square miles. How many 
square miles are there in these six states together ? 

8. In the year 1880 the number of people in Alaska was 
30,000; in Arizona, 40,440; in Dakota, 135,177; and in 
Idaho, 32,610. How many people altogether were there in 
these four territories ? 

9. Find the united length of the four following rivers : 
Mississippi, 3,160 miles; Missouri, 3,100 miles; Mackenzie, 
2,420 miles ; and Yukon, 2,240 miles. 

10. What is the number, from which if 10,739 be taken, 
the result is 39261 ? 



136 SUBTRACTION. 



SUBTRACTION. 

1. Edith had 35 cents in her bank ; how many cents did 
she have in the bank after taking out 8 cents ? 8 cents from 
35 cents leave how many cents ? 

2. Mr. Frank cut 5 yards from a piece of muslin containing 
29 yards ; how many yards were left in the piece ? 

293. The difference between two numbers is a number 
which, added to the smaller, will give a result equal to the 
greater. 

Thus, the difference between 15 and 8 is 7, because 7 added to 8 
makes 15. 

294. Subtraction is the process of finding the difference 
between two numbers. 

295. The greater of two numbers whose difference is to 
be found is called the minuend. 

296. The smaller of two numbers whose difference is to 
be found is called the subtrahend. 

297. The result in subtraction is called the remainder. 

298. The sign of subtraction is — , and is read minus 
or less. 

Thus, 14—6 means that 6 is to be subtracted from 14, and may be 
read, 14 mimes 6, or 14 less 6. 

1. Can you subtract 7 cents from 19 cts. ? What kind of 
numbers are they ? 

2. Can 8 marbles be subtracted from 12 eggs ? What 
kind of numbers are they ? 

299. Principles. — I. Subtraction can be performed 
only between like numbers, and units of the same order. 

II. The remainder is always like the minuend and 
subtrahend. 



SUBTRACTION-. 137 

30(h To find the difference between numbers ex- 
pressed by tivo or more figures. 

1. Eequired the difference between 20,951 and 14,785. 

Process. Explanation. — 1. Begin with units. 

r\ r\ n r~ ® cannot be subtracted from 1 ; 1 ten 

sd (J . U O 7 is therefore taken from the 5 tens of 

/ / ty Q £\ ^ ne mmuen d 5 changed to units and 

^ *4v i O O added to the 1, making 11 units ; 5 

~Z T - "7 units subtracted from 11 units leave 6 

•/ U A.71S. units, which is written in units' place in 

the remainder. 

2. Taking 1 ten from the 5 tens of the minuend left 4 tens ; the 8 
tens of the subtrahend cannot be subtracted from 4 tens, and hence 1 of 
the 9 hundreds of the minuend is taken and reduced to tens, making 14 
tens in all; 8 tens from 14 tens leave 6 tens, which is written in tens' 
place in the remainder. 

3. 7 hundreds subtracted from the 8 hundreds left after reducing 1 of 
the hundreds to tens, leave 1 hundred, which is written in hundreds' 
place in the remainder. 

4. Since there are no thousands in the minuend, 1 ten-thousand is re- 
duced to thousands, and the 4 thousands of the subtrahend subtracted 
from the 10 thousands leave 6 thousands for thousands' place in the 
remainder. 

5. From the 1 ten-thousand left in the minuend the 1 ten-thousand 
is subtracted, leaving ten-thousands for the highest order in the re- 
mainder. 

Note. — In practice the following is all that needs to be said or 
thought: 5 from 11, 6; 8 from 14, 6; 7 from 8, 1; 4 from 10, 6; 1 from 
1,0. 

II. Eequired to find the difference between 3,000 and 1,357. 

Process. Explanation. — Since there are no hun- 

dreds, tens, or units in the minuend from 

zj (J (J (J which to subtract the corresponding orders 

/ Q £\ 'y °^ * ne su btrahend, 1 of the thousands is 

^ ) *3 ^ ' taken and changed to hundreds, 1 of the 10 

/ hundreds thus obtained is changed to tens, 

/ } J^3 Arts. and 1 of the 10 tens is reduced to 10 units. 

7 units from 10 units leave 3 units, 5 tens 

from 9 tens leave 4 tens, 3 hundreds from 9 hundreds leave 6 hundreds, 

and 1 thousand from 2 thousands leaves 1 thousand. 



188 SUBTRACTION. 

301. Rule. — I. Write the subtrahend under the 
minuend, so that units of the same order shall stand 
in the same column. 

II. Begin at the right and subtract each figure of the 
subtrahend from the corresponding figure of the min- 
uend. 

III. When any figure of the minuend is less than that 
of the same order in the subtrahend, add 10 to it before 
subtracting ; and then consider the next figure of the 
minuend as diminished by 1. 

Proof. — Add the remainder to the subtrahend, and 
if the sum is equal to the minuend, the irorh is correct. 

302. Write from dictation anil find the differences : 



1. 


2. 


3. 


4. 


$678.53 


$563.48 


$3,640.52 


$1,758.30 


345.21 


276.59 


1,828.39 


864.58 


5. 


6. 


7. 


8. 


$1,000 


$1,005 


$80.09 


$40.50 


948 


406 


79.10 


35.75 


9. 


10. 


n. 


12. 


200,830 


160,087 


100,010 


800,000 


154,376 


83,548 


27,645 


675,304 


13. 


14. 


15. 


16. 


160,000 


70,700 


50,000 


100,000 


89,100 


69,690 


38,070 


99,099 



303. The following exercise, which requires pupils to give 
remainders at sight, is designed to facilitate written work in 
subtraction, 



SUBTRACTION'. 139 

304. Announce remainders at sight: 

Read answers : 1. From left to right ; 2. From right to left ; 3. From 
top to bottom ; 4. From bottom to top ; 5. As directed by the teacher. 





1st. 


2d. 


3d. 


4th. 


5th. 


6th. 


7th. 


8th. 


9th 




(41 


41 


41 


41 


41 


41 


41 


41 


41 


A 


u 


2 


3 


4 


5 


6 


7 


8 


9 




(42 


42 


42 


42 


42 


42 


42 


42 


42 


B 


2 


3 


4 





6 


rv 
i 


8 


9 


10 



(43 43 43 43 43 43 43 43 50 

C ) 3 4 5 6 7 8 9 10 10 



^44 44 44 44 44 44 44 49 49 

^ \A A A J7 _8 _9 10 _9 10 

^45 45 45 45 45 45 43 48 48 

E \A A A A A 1° A A A 



F 



^ 46 46 46 46 46 47 47 47 47 

\A A A A A A A A 10 



Note. — The attainment of rapidity should be regarded as of primary 
importance. 

305. The preceding exercise is a model for others which 
should be made by substituting for 41, 42, etc., 51, 52, or 
61, 62, etc. 

306. When several numbers are connected by the signs 
+ and — , the operations indicated must be performed in the 
order of the signs, commencing at the left. 

Thus, to find the value of 6 + 9-5 + 7, add 6 and 9, subtract 5 from 
the sum, and add 7 to the remainder. 



140 SUBTRACTION. 

307. The parenthesis, ( ), is used to include several 
quantities to be treated as one. A vinculum, ~ , serves 
the same purpose. 

Thus, (6 + 9)— (5 + 7) denotes the difference between the sum of 6 and 
9 and the sum of 5 and 7. The same is denoted by 6 + 9—5 + 7. 

308. Find the value of: 

i. 14—7 — 3 + 4. 5. 100 — (25 + 42) + 17. 

2. 14— (7 — 3) + 4. 6. 452 — 127 + 231. 

s. 14— 7— (3 + 4). 7. 600—40 + 100 + 10. 

4 . 36 + 12 — (9 + 7). s. 996 — (325-140 — 170). 

ORAL PROBLEMS. 

309. i. If blue stone is sold for 65 cents a foot at one 
quarry and 8 cents a foot less at another quarry, for how 
much a foot is it sold at the second quarry ? 

2. If shingles are sold one month for 92 cents a bunch and 
the next month at 6 cents less, for how much a bunch are 
they sold the second month ? 

3. The newels for a front stoop cost 
$46 and for the front fence $38. How 
much less did the newels for the fence 
cost ? 

Jf. If Mr. Jefferson's residence was 
48 feet high, how high was the top 
of his flag pole, which extended 9 feet 
above the top of his residence ? 

5. How long is a boat that is 50 feet shorter than one 93 
feet in length ? 

6. If a bushel of potatoes weighs 58 pounds and a bushel 
of apples 28 pounds, how much more do the potatoes weigh ? 

7. John stopped working at 12 o'clock noon ; he had been 
working five hours. When did he begin ? 

8. Frances had 22 cents and spent 8 cents for car fare. 
How many cents did she then have ? 




SUBTRACTION. 141 

WRITTEN PROBLEMS. 

310. 1. In the year 1880 the cotton crop of Alabama 
was 700,000 bales; that of Mississippi, 956,000. Which 
crop was the greater, and how much ? 

2. The cotton crop of the United States in 1880 amounted 
to 5,737,000 bales. Mississippi, the largest cotton-growing 
state, raised 956,000 bales. How many bales did the other 
states raise ? 

3. If the iron product of all the countries of the world 
in one year was 21,000,000 tons ; and that of Great Britain, 
8,490,000 tons, what was the iron product of all the other 
countries ? 

J±. Find the difference in the heights of Mt. Washington, 
6,293 feet, and Mt. Marcy, 5,344 feet. 

5. Find the difference between the lengths of the river 
Rhine, 809 miles, and that of the Mississippi, 3,160 miles. 

6. If the diameter or line passing through the center of 
Jupiter is 85,300 miles long, and the diameter of the Earth 
7,912 miles long, how much longer is the diameter of 
Jupiter ? 

7 . If a silk dress cost $41.75, and a cashmere dress, $22.50, 
how much more did the silk dress cost ? 

8. Mr. Moore had $65.15 in bank. How much did he 
have in bank after, drawing out $46.50 ? 

9. Tim bought newspapers for $1.25 and sold them for 
$1.65. How much did he make ? 

10. Mr. Jordan's house and lot cost him $6000 ; he spent 
$500 in repairing the house. How much did he lose on sell- 
ing them for $4750 ? 

11. Frank earned $7 a week and paid out $5.50 for his 
expenses. How much did he have left ? 

12. In one piece of cloth there are 72 yards ; in another, 
58 yards. The first piece contains how many more yards 
than the second ? 



142 MULTIPLICATION. --*• ■ 

MULTIPLICATION. 

• 

1. Carl earned $5 a week for 4 weeks. How much did he 
earn altogether ? Four 5's are how many ? Four times five 
equals how many ? 

2. Jane made 4 aprons, each out of 2 yards of calico. How 
many yards of calico did she use ? 

311. Multiplication is the process of taking one number 
as many times as there are units in another. 

312. The number repeated or multiplied is called the 
multiplicand. 

313. The number which indicates how many times the 
multiplicand is to be taken is called the multiplier. 

314. The result obtained by multiplication is called the 
product. 

315. The factors of a number are the numbers which 
multiplied together will produce it. 

The multiplicand and multiplier are factors of the product. 

316. When the product of more than two numbers is 
found by multiplying the first by the second, the result by 
the third, etc., the process is called continuous multiplica- 
tion, and the final result the continued product. 

1. 4 times 5 are how many ? 5 times 4 are how many ? 
What is the continued product of 2, 3 and 4 ? Of 4, 3 and 
2 ? Of 4, 2 and 3 ? 

317. The product of two or more factors is the same in 
whatever order they are used. 

318. The sign of multiplication is x , and is read mul- 
tiplied hy, or times. 

Thus, $7 x 5 is read $7 multiplied by 5, or 5 times $7. 7x5, two ab- 
stract numbers, may be read, 7 times 5, or 5 times 7, 2 x 3 x 5 is read, 
the continued product of 2, 3 and 5, 



MUL TIP LIG A T10N. 



143 



1. Can you multiply $5 by 4 ? Is the multiplicand con- 
crete or abstract ? The multiplier ? The product ? 

2. Can you multiply 5 by 4 ? What kind of a number is 
the multiplicand ? The multiplier ? The product ? 

3. Can you multiply 5 by $4 ? $5 by $4 ? 

319. Principles. — I. The multiplicand may be either 
concrete or abstract. 

II. The multiplier is always abstract. 

III. The product is always like the multiplicand. 

320. 1. The last of the following exercises is a model for 
others, which should be constructed by substituting for 12 in 
their order the numbers smaller than 12, and changing the 
products accordingly. 

2. Another series of tables may be made by changing the 
order of the factors. 



321. Copy and commit to memory: 



5x 5= 25 


6x 6= 36 


7x 7= 49 


8x 8= 64 


5x 6= 30 


6x 7= 42 


7x 8= 56 


8x 9= 72 


5x 7 = 35 


6x 8 = 48 


7x 9= 63 


8x10= 80 


5x 8= 40 


6x 9= 54 


7x10= 70 


8x11= 88 


5x 9= 45 


6x10= 60 


7x11= 77 


8x12= 96 


5x10= 50 


6x11= 66 


7x12= 84 




5x11= 55 


6x12= 72 






5x12= 60 








9x 9= 81 


10x10=100 


11x11=121 


12x12=144 


9x10= 90 


10x11=110 


11x12=132 




9x11= 99 


10x12=120 






9x12=108 








1x12=12 


4x12=48 7x12= 84 


10x12=120 


2x12=24 


5x12=60 8x12= 96 


11x12=132 


3x12=36 


6x12=72 9x12=108 


12x12=144 



144 MULTIPLICATION. 

322. Announce products at sight : 

Read answers : 1. Prom left to right ; 2. Prom right to left ; 3. From 
top to bottom ; 4. From bottom to top ; 5. In any order suggested by 
the teacher. 



A 



Jst. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 10th. 

j 6 8 10 95429 11 7 

ii J i J J li- J i i J. 

(6 53 12 11 9 865 7 

7454 11 7546 8 



54 10 12 94536 8 

111 7 7 3 2 4 10 6 8 9 



D 



( 9 11 12 8 10 11 4 9 12 7 

|H J i i5 ^ J ^ J-'i 



G 



6 



\ 11 3 7 10 11 4 11 4 3 4 

E \JL J. 1°. A I? 12 ' _i _§ ' ■*? _? 

6 8 2 8 9 6 10 12 7 4 

5 11 10 3 5 11 8 6 3 6 



^85357 11 12 65 9 

j_8 _5 tt ' _7 9 12^J?^ _? 

j 3 6 2 8 12 10 8 10 9 11 

H |_8 10 9 6 12 10 4 _5 _8 _8 

, 2 4 5 12 11 7 11 12 2 6 

M2 5 8 11 7 12 5 10 11 12 



, 3 12 8 10 7 9 7 12 9 10 
Ml 9 12 9 7 12 4 8 10 12 



MULTIPLICA TION. 



145 



323. To find the product of two numbers when the 
multiplier is expressed by two or more figures. 

I. Required the product of 274 multiplied by 125. 



rocess. 



274 

425 

J 370 
548 

27J, 

34,250 

Analysis. 
274 x 5 = 1,370 
274 X 20 = 5,480 
274 X 100 = 27,400 
274 x 125 = 84,250 



Explanation.— 1. The multi- 
plicand is multiplied by the 5 
units of the multiplier, giving for 
a product 1370, which is so written 
that the right-hand figure shall 
stand under the units' figure of the 
multiplier. 

2. The multiplicand is multi- 
plied by the 2 tens of the multi- 
plier, giving for a product 548 
tens, which is so written that the 
right-hand figure shall stand 
under the tens of the multiplier. 

3. In the same manner, the 
multiplicand is multiplied by the 
1 hundred of the multiplier. 

If. The partial products are 
added, giving 34,250 for the en- 
tire product. 



324. Rule. — I. Write the multiplicand and multi- 
plier so that units of the same order shall stand in the 
same column. 

II. Multiply the multiplicand by each significant 
figure of the multiplier, writing the first or right-hand 
figure of each partial product under the figure used 
as a multiplier. 

III. Add the partial products. 

Proof.— Multiply the multiplier by the multiplicand, 
and if the product is the same, the work is correct. 



Note. — For the multiplication of two numbers where one of the 
factors contains a fraction, see Art, 237* 



146 MULTIPLICATION. 

32 5 • Find the products : 

i. 234 x 35. 5. 684 x 56. 9. 1098 x 75. 

2. 456 x 43. o. 875 x 47. 10. 2865 x 68. 

3. 542 x 52. 7. 739 x 65. u 3709 x 79. 

4. 278 x 34. *. 927 x 46. i* 4087 x 93. 

326. To /£wd f/^6 product of two factors, when there 
are ciphers at the right of the significant figures of one 
or both. 

I. Eequired to multiply 123 x 100. 

Process. Explanation. — 123 multiplied by 100 

is equal to 123 times 100, or 123 hundreds. 
7 & J Hence, to multiply a number by 10, 100, 

1000, etc., annex as many ciphers to the 
multiplicand as there are ciphers in the 
multiplier. 



400 



42,300 

II. Required to multiply 230 by 1600. 

Process. Explanation. — 230 multiplied by 1600 

equals 23 x 10 x 16 x 100 ; and since the 

d cj (J factors may be taken in any order, 23 may 

. / be multiplied by 16 and the product by 

4 0OO 10x100, or 1000. Hence, to find the 

product of two factors, one or both hav- 

438 ing ciphers at the right of the significant 

figures, first find the product of the num- 

sd zJ bers expressed by the figures at the left of 

the ciphers, and then annex to it as many 



A O f) f) f) ciphers as there are at the right in both 

. ^ of the factors. 

327. Rule. — Find the -product of the -parts of the 
multiplicand and multiplier expressed by the signifi- 
cant figures at the left of the ciphers, and then annex 
as many ciphers as there are at the right in both factors, 



MULTIPLICATION. 147 

338. Find the products : 

1. 321 x 30. 4. 480 x 60. 7. 6800 x 400. 

2. 456 x 60. 5. 590 x 70. s. 5960 x 600. 
.?. 748x50. 6. 880x90. 9. 8700x580. 

329. Find the continued products : 

1. 693 x 30 x 40. 4. 987 x 60 x 80. 7. 5000 x 20 x 8. 

2. 478 x 40 x 50. s. 800 x 46 x 70. 8. 4040 x 40 x 30. 

3. 600x56x30. o. 650x30x25. 9. 3080 x 9x60. 

330. When the sign of multiplication is used with the 
signs of addition and subtraction, the indicated products 
must be found first, except when the quantities joined by the 
latter are united by the parenthesis or vinculum. 

Thus, to find the value of 5 + 4x3—8, multiply 4 by 3, add the 
product to 5, and subtract 8 from the sum. 

331. Perform the operations indicated : 



l. 874x35 — 430x60. 4. 400 — 64x375 + 25. 



2 , 1076 — 428x670. 5. 60x50 + 40x30. 

3. 145 + 870x65. c>. 700 -(400 + 35) +80. 

ORAL PROBLEMS. 

332. 1. At $1^ a load what will 14 loads of sand cost ? 

2. If there are 13 cubic feet in one load of sand how many 
cubic feet are there in 3 loads ? 

3. If one foot of fence costs $2±, how much will 8 feet of 
fence cost ? 

Jf. A foot of zinc cornice is worth $H. How much are 20 
feet worth ? 

5. In building a house each window was found to cost $5. 
How many dollars did 9 windows cost ? 

6. At $8 a window, how much would 9 windows cost ? 

7. At 3 cents a foot, how much will 12 feet of flooring cost ? 



148 MUL TIP LIG A TION. 

8. If each of 7 rose leaves has 9 leaflets, how many leaflets 
have the 7 rose leaves together ? 

9. If laths sell for $3 per thousand, how much will 12 
thousand cost ? 

WRITTEN PROBLEMS. 

333. i. In ten years a builder erected 60 houses, using 
120 thousand bricks for each house. How many bricks alto- 
gether did he use ? How many for 35 houses ? 

B . What will be the cost of 60 x 120 
thousand bricks, at $6 a thousand ? 

3. If each of 60 houses costs $4000 and 
is sold for $5800, how much money is made 
by the sale of the 60 houses ? 

Jf. In building a Chinese pagoda or tem- 
ple, 425 men worked 395 days of 10 hours 
each. How many hours^ labor were required ? 
5. If the beams for houses cost $20 a 
thousand feet, how much will 35^ thousand 
feet cost ? 

6. One barrel of flour weighs 196 pounds. How many 
pounds do 196 barrels of flour weigh ? 

7. Every commodore in the U. S. navy receives $5000 a 
year. How much do 25 commodores receive ? 

8. Every captain in the U. S. navy receives $4,500 a year. 
How much do 50 captains receive ? 

9. What will be the cost of 58 rolls of carpet, each con- 
taining 74 yards, at $2 a yard ? 

10. If I buy 80 barrels of flour at $5.60 a barrel and sell 
them for $476, do I make or lose, and how much ? 

11. There are 60 minutes in one hour ; how many minutes 
are there in 24 hours or one day ? 

12. There are 60 seconds in a minute ; how many seconds 
are there in one day ? 

13. A man earns $2 per hour. How much does he earn in 
5 years of 200 working days each, working 8 hours each day ? 




division. 149 



DIVISION. 

1. If 8 equal rows of desks seat 40 pupils, how many pupils 
does each row seat ? Forty divided into eight equal parts 
give how many to each part ? 

2. $27 were paid for 9 yards of silk. What was the cost 
of one yard ? 

3. How long will it take James to walk 36 miles, if he 
walks 4 miles an hour ? 

334. Division is the process of finding how many times 
one number is contained in another of the same kind ; or of 
dividing a number into equal parts, the number of parts being 
given. 

335. The number divided is called the dividend. 

336. The number by which the dividend is divided is 
called the divisor. 

337. The result is called the quotient. 

The quotient indicates the number of times one number is contained 
in another; or it is one of the equal parts into which a number is divided. 

338. The remainder is the part of the dividend left 
over when the division is not exact. 

Thus. $8 is contained 6 times in $55 and $7 are left over; $7 in this 
case are the remainder. The remainder is a part of the dividend, and 
therefore of the same denomination as the dividend. 

339. The sign of division is -^-, and is read divided by. 
Division is also indicated by writing the dividend above and 
the divisor below a short horizontal line, or by writing the 
divisor before the dividend with a curved line between. 

45 
Thus, each of the expressions, 45-f-9, — , and 9) 45 means that 45 is 

to be divided by 9. 



150 



DIVISION. 



340. i. The signs of multiplication and division take 
precedence over those of addition and subtraction, except 
when the quantities joined by the latter are united by the 
parenthesis or vinculum. Thus, 

14-3 x 2 + 8-4=14-6 + 2=8 + 2 = 10. 
(14-3) x 2 + 8-5-4=11 x 2 + 8-4=22 + 2=24. 

2. When the signs of multiplication and division are used in 
immediate succession, the parenthesis or vinculum is needed 
to indicate which operation is to be performed first. 

Thus, 5x8 — 4x2 is indefinite. It may mean (5 x 8) -J- (4 x 2), or 
(5 x 8 - 4) x 2, or 5 x (8 - 4 x 2), etc. 



341. i. The last of the following exercises is a model for 
others which should be constructed by substituting for 12 in 
their order, the numbers smaller than 12, and changing the 
dividends accordingly. 

2. Another series of tables may be made by using the 
quotients for divisors. 

342. Copy and commit to memory : 



25+5= 5 


36-6= 6 


49-7= 7 


64-8= 8 


30-5 = 6 


42-6= 7 


56-7= 8 


72-8= 9 


35-5 = 7 


48-6= 8 


63 - 7= 9 


80 - 8=10 


40-5 = 8 


54+6= 9 


70- 7 = 10 


88 - 8=11 


45-5 = 9 


60 - 6=10 


77 - 7 = 11 


96 - 8=12 


50-5=10 


G6+ 6=11 


84 - 7 = 12 




55-5=11 


72 - 6=12 






60-5=12 








81-9 = 9 


100-10=10 


121-11=11 


144-12=12 


90-9=10 


110-10=11 


132-11=12 




99-9=11 


120-10=12 






108-9=12 








12-1=12 


48-4=12 84-7=12 120-10=12 


24-2=12 


60-5=12 96-8 = 12 132-11 = 12 


36-5-3=12 


72-6=12 108-9=12 144-12=12 



DIVISION. 151 

343. Accuracy and rapidity in division are determined 
largely by the pupil's ability to give quotients at sight. The 
following exercise should be frequently practiced. 

344. Announce quotients at sight: 

Read in different directions, as directed by the teacher, naming only 
the quotients. 

1st. 2d. 3d. 4th. 5th. 6th. 7th. 
A { 8)80 6)42 4 )28 6)30 8 )48 4 )12 3)24 

B j 6)54 8)_64 9 )_45 8 )_24 5 )_20 4)24 6)18 

C j 4)16 6)30 8)32 9 )_54 2 )_18 6 )_36 3)27 

p j 5)45 7)56 3)21 2)16 8)40 5)25 9)27 

E \ 7)14 6)48 3)12 5)15 2)14 6)6 4)32 

F 



G 
H 

I 
J 
K 
L, 
M 

















3)3 


2)12 


9)72 


8)16 


7)63 


5)40 


7)7 


4)36 


3)9 


8)16 


9)36 


2)10 


6)12 


3)15 


9)9 


5)20 


7)35 


8)72 


9)18 


5)5 


4)8 


3)15 


6)24 


8)56 


7)28 


5)35 


3)12 


7)21 


7)49 


9)63 


5)10 


4)20 


7)42 


3)18 


9)81 


10)50 


10)100 


11)77 


11)99 


10)80 


11)66 


10)90 


10)60 


11)110 


11)121 


12)120 


12)108 


10)70 


11)88 


11)55 


12)132 


12)144 


12)60 


12)96 


12)72 


12)84 



152 DIVISION. 

345. In the preceding exercise the divisions are exact. In 
practice there is usually a remainder, and the following exer- 
cise is therefore given as a model for a series of exercises, 
the full series to be constructed by using successively each of 
the other digits in place of 9. The exercises may be read 
from the book, or written by the pupils on slates or papers, 
or arranged on the blackboard by the teacher or pupil. 

346. Announce at sight quotients and remainders: 

Read in different directions, naming only quotients and remainders. 

1st. 2d. 3d. 4th. 5th. 6th. 7th. 
9)56 9)31 9)46 9)83 9)95 9)11 9)22 



A 
B 

C 
D 
E 
F 

a 

H 

I 
J 
K 



9)13 9)82 9 )71 9)10 9)70 9 [91 9)40 

9 [93 9 [53 9 [41 9)80 9 )J.6 9^21 9)65 

9)J.2 9)79 9)42 9)66 9)69 9)44 9)33 

9 )20 9)14 9)67 9)92 9 [17 9 [50 9)24 

9)57 9)19 9)89 9 [15 9)23 9)61 9)85 

9)25 9 [49 9 [38 9 [8 8 9 [59 9 [29 9)94 

9 )68 9)55 9)43 9)73 9)86 9)32 9)62 

9)48 9)77 9)35 9)75 9)52 9)26 9 [74 

9 [39 9)87 9 [30 9 [37 9 )_58 9 [34 9)84 

j 9 [60 9 [78 9)51 9 [64 9 [76 9 [48 9 [2 8 



DIVISION. 



153 



347. To find the quotient of one number divided by 
another "when the divisor is not greater than 12. 

I. Kequired the quotient of 8,551 divided by 6. 



6 



Process. 

8,554 

4 ,^25-^™. 



Analysis. 

6000 + 6 — 

2JfiO + 6 = 

120 + 6 = 

30 + 6 = 

1 + 6 — 



1000 
JfiO 

20 
5 



8,551 + 6 = 1,425£ 



Explanation. — 1. The 8 thou- 
sands of the dividend divided by 6 
give a quotient of 1 thousand and 
a remainder of 2 thousands or 20 
hundreds. The 1 thousand is writ- 
ten in thousands' place in the quo- 
tient, and the 20 hundreds added 
to the 5 hundreds for the next par- 
tial dividend. 

2. The 25 hundreds divided by 6 
give a quotient of 4 hundreds and 
a remainder of 1 hundred or 10 
tens. The 4 hundreds are written 
in hundreds' place in the quotient 
and the 10 tens added to the 5 tens 
for the next partial dividend. 

3. The 15 tens divided by 6 give a quotient of 2 tens and a remainder 
of 3 tens or 30 units. The 2 tens are written in tens' place in the quotient 
and the 30 units added to the 1 unit give 31 for the last partial divi- 
dend. 

4. The 31 units divided by 6 give a quotient of 5 units and a 
remainder of 1 unit. The 5 is written in units' place in the quotient 
and the remainder, written over the divisor to indicate the division, 
is annexed to the quotient. 

Note. — In practice, the different orders of the dividend are regarded 
as units and the division is made as follows : 6 in 8 once ; in 25 four 
times; in 15 two times; in 31 five times with a remainder of 1, which 
is written over the dividend and annexed to the quotient. 

348. Find the quotients : 



i. 4 ) 378504. 

2. 6 ) 503976. 

3. 8 ) 575632. 



6. 3)579684. n. 2 ) 579375. 



4. 10 ) 75843 6. 

5. 12 ) 628920. 



7. 5 ) 784321 

9 



12. 12 ) 763896. 



7 ) 910028. is. 11 ) 893472. 



9 ) 551079. 14. 10 ) 57840 5. 



io. 11 ) 678125. is, 9 ) 82897 6. 



154 DIVISION. 

349. To find the quotient when the divisor contains 
tivo or more figures. 

I. Kequired the quotient of 24,989 divided by 31. 

Process. Explanation. — 1. 

The least number of 



O 4 ) &4<, y O y ^ O (/ (/ 31 left-hand figures that 

24,8 



I8(j 




486 




3 




Analysis. 




24,800 -f SI = 


800 


186 -+ 31 = 


6 


8 -T- $1 = 


3 

3 i 



will contain the divisor 
is three, and hence the 
first partial dividend is 
249 hundreds. 

2. 31 is contained in 
249 hundreds, 8 hun- 
dred times with a re- 
mainder. The 8 is writ- 
ten in hundreds' place 
in the quotient, the 
divisor multiplied by 
it, and the product sub- 
tracted from the par- 
tial dividend. 

3. To the remainder, 
24,989 ~ 81 — 806/ 7 i hundred, is annexed 

the 8 tens of the divi- 
dend, giving 18 tens for the second partial dividend. 31 is not contained 
in 18 ; a cipher is therefore written in the quotient and the 9 units of the 
dividend annexed to the 18 tens, giving 189 units for the third partial 
dividend. 31 is contained in 189 units, 6 times with a remainder. The 
6 is written in units' place in the quotient, the divisor multiplied by it, 
the product subtracted, and the remainder written over the divisor and 
annexed to the quotient. 

350. When the operation of division is fully expressed, as 
above, the process is called long division ; where a part of 
the operation is performed mentally, as in Art. 347, the proc- 
ess is called short division. 

351. Rule for Long Division. — I. Write the divisor 
at the left of the dividend with a curved line between. 



DIVISION. 155 

II. Use as the first paHial dividend the fewest left- 
hand figures of the dividend that will contain the 
divisor. Divide, and write the quotient at the right of 
the dividend with a curved line between. 

III. Multiply the divisor by this quotient figure, sub- 
tract the result from the partial dividend, and to the 
remainder annex the next figure of the dividend. 

IV. Divide the number thus obtained by the divisor, 
write the result in the quotient, multiply the divisor by 
it, and subtract as before. 

V. So continue until all the figures of the dividend 
have been annexed. 

VI. Should any partial dividend not contain the 
divisor, place a cipher in the quotient, annex another 
figure to the partial dividend, and proceed as before. 

Proof. — Multiply the divisor and quotient together 
and to the product add the remainder ; if the work is 
correct the sum will equal the dividend. 

352. Find the quotients : 

1. 2078-^-13. 7. 10086-^-55. is. 73002-4-430. 

2. 4065-^33. s. 200944-98. 14. 650844-560. 

3. 55734-48. 9. 304764-45. 15. 543724-452. 

4. 48914-37. io. 865204-91. ». 476524-563. 
*." 6254-4-41. *i- 670084-76. 17. 38791 — 672. 
6. 73094-42. 12. 543214-55. is. 290004-586. 

ORAL PROBLEMS. 

353. i. A crystal of snow has 6 points or angles. How 
many crystals will together have 54 angles ? 

2. One oak leaf has 8 lobes or parts. How many leaves 
together have 72 lobes ? 

3. If 3 small loads make a cubic yard of sand, how many 
loads will make 12 cubic yards ? 



156 DIVISION. 

Jj,. If one square foot of flagging costs 30 cents, how many 
square feet can be bought for 90 cents ? 

5. How many thousand feet of rafters can be bought for 
$80, when one thousand costs $20 ? 

6. A silk neck-tie costs 96 cents ; a cotton one costs one- 
twelfth as much. How much does the cotton tie cost ? 

7. 12 large buttons cost $1.32 ; how much did each button 
cost ? 

WRITTEN PROBLEMS. 

354. 1. If the general of the U. S. army receives $324,000 
for 24 years of service, how much does he receive for each 
year ? 

2. 36 colonels in the U. S. army receive $126,000 in one 
year. How much does each colonel receive ? 

3, A bushel of peaches weighs 28 pounds. How many 
bushels weigh 3460 pounds ? 

Jj,. A bushel of Indian corn weighs 56 pounds. How many 
bushels weigh 40360 pounds ? 

5. If a steamship sails 
18 miles per hour, how 
long will it be in sailing 
3700 miles ? 

6. If a steamship uses 
42 tons of coal in one day, 
how long will it be in using 
1000 tons ? 

7. If 120 thousand bricks 
are required for building each house, how many houses can 
be built with 600000 bricks ? 

8. San Francisco is 3302 miles distant from New York ; 
Philadelphia is only 88 miles distant from New York. How 
many times as far away is San Francisco ? 

9. The President o* the United States receives $50,000 a 
year. How much does he receive per week, allowing 52 
weeks to the year ? 




MISCELLANEOUS PROBLEMS 157 

10. How much will it cost to clothe an army of 16800 men 
at $23 for each man ? 

11. If a man's income is $5,000,000 a year, what is it per 
day, allowing 365 days to the year ? 

MISCELLANEOUS ORAL PROBLEMS. 

355. 1. What will 3^ yards of calico cost at 12 cents a 

yard ? 

JL How much will 4 yards of muslin cost at 12^ cents a 
yard ? 

3. If 5 boys can do a piece of work in 12 hours, how long 
will it take one boy ? 

Jf. If one boy can do a piece of work in 60 hours, how long 
will it take 12 boys ? 

5. If a river flows 35 miles in 7 hours, how far will it flow 
in one hour ? In 6 hours ? 

6. Edward earns $16 in 4 weeks, how much can he earn in 
5 weeks ? In 9 weeks ? 

7. How much will Mr. Green earn in 8 weeks if he earns 
$36 in 3 weeks ? 

8. The smaller of two numbers is 14 ; the difference be- 
tween them is 9. What is the greater number ? 

9. After $7 were spent from a purse, there were $44 left in 
it ; how much did the purse contain at first ? 

10. The sum of two numbers is 25 ; one of the numbers 
is 8. What is the other number ? 

11. Three numbers added together equal 48 ; two of the 
numbers are 8 and 10. What is the third number ? 

12. If the product of two numbers is 72 and 8 is one of 
the numbers, what is the other ? 

13. What number divided by 6 will give 9 for a quotient ? 
14- What number multiplied by 7 will give 63 for a 

product ? 

15. If 8 copy books cost 96 cents, what will 5 cost ? 



158 MISCELLANEOUS PBOBLEMS. 

16. If 6 slates can be bought for 42 cents, for how much 
can one slate be bought ? 3 slates ? 

17. How many bottles of ink can be bought for 40 cents, 
if 3 bottles can be bought for 24 cents ? 



MISCELLANEOUS WRITTEN PROBLEMS. 

356. i. The difference between two numbers is 1378 ; the 
less number is 4621. Find the greater number. 

2. One of two factors is 843 ; the product is 384,408. Find 
the other factor. 

3. The sum of three numbers is 68,703 ; two of the num- 
bers are 15,960 and 28,054. Find the third number. 

Jj,. If a divisor is 728 and the quotient 316, what is the 
dividend ? 

5. How much will 72 yards of silk cost at $2£ per yard ? 

6 . Find the cost of 7 -J yards of ribbon at 4 cents a yard. 

7. If Mr. George buys 40 barrels of flour at $4|- a barrel 
and sells them so as to make $75, for how much does he sell 
the 40 barrels ? 

8. Frank earned $300 a year ; at the same rate how much 
would he earn in 3^ years ? 

9. Find the average price per pound of four qualities of 
sugar costing respectively 5, 7, 8, and 10 cents per pound. 

10. If 3 houses sell for the following amounts: $4,600, 
$5,400, and $8,000, what is the selling price of the three to- 
gether ? 

11. If 3 houses together sell for $18,000, for how much 
does each house sell ? What is the average selling price ? 

12. Mr. James saves $1200 per year ; Mr. Clare, 5 times 
as much as Mr. James ; and Mr. Howard twice as much as 
Mr. Clare. How much in all do they save per year ? 

13. Mrs. Pratt is 90 years old ; her daughter is ^ as old ; 
her granddaughter is \ as old as the daughter. How old is 
the granddaughter ? 



MISCELLANEOUS PROBLEMS. 159 

14. After spending $32, Matthew had $18 left out of his 
month's earnings ; how much did he earn in the month ? 

15. If 4 men require 36 days to do a piece of work, how 
many days will one man require ? 

16. If one man can do a piece of work in 144 days, in how 
many days can 9 men do the work ? 

17. 8 men dug a ditch in 27 days. How long would it 
have taken one man ? How long would it have taken 6 men ? 

18. In what time can 15 men build a wall which 5 men 
can build in 69 days ? 

19. If John earns $432 in 6 months, how much can he 
earn in 7 months ? In 9 months ? 

20. The dividend is 81,000 and the quotient, 375. What 
was the divisor ? 

21. What number multiplied by 33 will give 9,504 ? 

22. After paying $4,375 towards purchasing a house, 
Mr. Brown left $5,625 of the price unpaid. For how much 
did he buy the house ? 

23. What will be the cost of 63 bushels of wheat at $1.12 
per bushel ? 

24. If screen doors sell for $4.25 each, for how much will 
5 screen doors sell ? 

25. If 5 equal pipes can empty a cistern in 65 minutes, 
how long will it take 4 of the pipes ? 

26. A merchant made $3000 by selling 7000 barrels of 
flour for $4.75 each. How much did the 7000 barrels 
cost him ? 

27. The product of three numbers is 33,005 ; two of the 
numbers are 35 and 41. Find the third number. 

28. If 65 tons of coal are given in exchange for 25 tons 
of hay worth $13 a ton, at how much per ton is the coal 
valued ? 

29. How much money will Mrs. Briggs have left out of 
four $10 bills and one $5 bill, if she buys 18 yards of silk at 
$2| per yard ? 



SECTION V. 



COMMON FRACTIONS. 




EQUAL PARTS OF UNITS. 



357. An integral unit is a whole or undivided unit. 
Thus, 1 apple, 1 orange, $1, 1, are integral units. 

358. A whole number, or integer, is an integral unit, 
or a collection of integral units. 

Thus, 5 apples, 4 oranges, $15, are integers. 

359. A fractional unit is one of the equal parts of a 
divided integral unit. 

Thus, \ of an orange, \ of an apple, -^ of a dollar are fractional units. 

360. A fractional number, or fraction, is a fractional 
unit, or a collection of fractional units. 

Thus, \ of a pound, f of a dollar, -|, f , are fractions. 

The word, fraction, comes from a Latin word meaning to break, and 
it stands for one or more of the parts obtained by breaking, cutting, or 
in any way dividing a unit into parts. 

361. A mixed number is a whole number and a fraction 
written together. 

Thus, 3J feet, 7| lb., 12 j, are mixed numbers, 



COMMON FB ACTIONS. 



161 



EQUAL PARTS OF UNITS. 
1. How many equal parts in each of the following : 




362. 1. To represent one half, or 1 of 2 equal parts, 
the figure 1 is written above the figure 2 with a line between 
them; thus, £ ; to represent two thirds, or 2 of 3 equal 
parts, the figure 2 is written above the figure 3 with a line 
between them ; thus, |. 

2. In the same manner : 



One fourth is written 


^ and stands for 1 of 4 


equal parts. 


One fifth " " 


1 a 

5 


a 


" 1 " 5 


?* » <■ 


Three fourths " . " 


3 ec 

4 


(C 


" 3 " 4 


** fc 


Four fifths " " 


4 (c 
5 


a 


«■ 4 " 5 


ft ft 


Five sevenths " " 


5 u 

7 


a 


« 5 « 7 


i.- u 



363. The number below the line is called the denomi- 
nator, and indicates the number of equal parts into which 
the unit is divided. The number above the line is called the 
numerator, and shows how many of the equal parts are 
taken. 

364, If the numerator of a fraction is less than its 
denominator, the fraction is called a proper fraction ; if the 
numerator is equal to or greater- than the denominator, the 
fraction is called an improper fraction. 

Thus, | and £ are proper fractions ; f and f are improper fractions. 



162 



COMMON FRACTIONS. 



365. Show by diagrams the meaning of the follow- 
ing : 

1 - i> 3> \> \y 6' TO' 3 ' ~5> ~6"> Toy J> Toy 10* 



2 3. 3 4 6 



4. 4, 



4 12 

8 y 9 y Jy 



3> ty Jy 1y t> To* *• 8> "5* 8> T* "§> f • 

366. Write from dictation the fractions in Art. 365. 

367. Write in words the fractions in Art. 365* 



ORAL PROBLEMS. 

368. i. In one unit how many halves are there ? 
units ? In 3 ? 

2. How many thirds are there in 
1 unit ? In 2 ? In 3 ? 

3. How many fourths, fifths, 
sixths, are there in 1 ? In 2 ? 
In 3? 



In 2 



in 



12 v 
3 • 



Jj,. How many units are there in § ? 
How many unit! 
In -V ? In J3 8 - ? 
How many units 




are there in J ? In f ? In \ 2 - 
In \°- ? In 2 X 4 - ? 



11 10 - 



6. 

v? 

7. In -fjy how many units are trier. . 

8. In I how many units and thirds are there ? In 

9. How many units and fourths are there in f ? 

10. In 1 unit how many halves are there ? I 
fourths ? 



Inf? 
? In 

? In 



In -^ how many units are there ? 
In 4 how manv units an 



±u 10 
3 ■ 



In 
How 



4 • 

many 



COMMON FBACTIONS. 



163 



I? 



I? 



Ninths ? 



11. In one half how many fourths are there ? In f ? In 

|? 

12. How many thirds are there in 1 unit ? How many 
sixths ? 

13. One third is equal to how many sixths ? § ? 
1^. One unit is equal to how many ninths ? 
15. One third is equal to how many ninths ? § 
iff. One third equals how many sixths ? 

Twelfths ? 

J7. One half equals how many fourths ? 
Twelfths ? 

i£. One fourth equals how many eighths ? 
One sixth equals how many twelfths ? 
f equals f ; how many thirds equal f ? 
f equal £ ; f equal how many halves ? 
Two fourths equal how many 



Eighths 



Twelfths ? 



4 ? 

6 " 

.4 V 
4 • 



A V 



6. V 
4 * 



I? 



■ ! ^^T ■"■■■ ^™ 

- f '* ! ' \ ■ 
I 



m 

21. 

22. 
eighths ? 

i?c?. Two fourths equal how many 
twelfths ? 

2Jf. Three fourths equal how many 
eighths ? How many twelfths ? 

25. One third equals how many sixths ? 

26. Two thirds equal how many sixths ? 

27. One third equals how many twelfths ? Two thirds ? 

28. One fourth equals how many twelfths ? 

i?#. Two fourths equal how many twelfths ? Three fourths ? 
30. One equals how many tenths ? 
c?i. One half equals how many tenths ? 

32. One fifth equals how many tenths ? Two fifths ? 

33. To what like parts may halves and fourths be reduced ? 
Halves, fourths, and eighths ? Halves, thirds, and sixths ? 
Halves, fifths, and tenths ? Thirds, fourths, and sixths ? 
Thirds, sixths and ninths ? Thirds and fifths ? Fourths 
and fifths ? 



164 COMMON FRACTIONS. 



ADDITION OF FRACTIONS. 

i. If William pay ^ of a dollar for fire crackers and f of a 
dollar for Eoman candles, how much does he pay for both ? 
2. What is the sum of \ and \ ? % and f ? f and f ? 

369. Like parts of the same or of equal units are added 
by finding the sum of the numerators. 

Thus, J + f + } = f = If = H. 

370. i. In order to add unlike parts of the same or of equal 
units, such as halves and fourths, it is necessary to reduce 
them to like parts. Thus, 

1. To find the sum of ^ and J, the one half is reduced to fourths. 

1 — 2. 2 i 1 — 3 

^ — 1~ > ¥ + ¥ — ¥• 

2. To find the sum of f and f , the two thirds are reduced to sixths. 

2 — 4. 4_i_5._9_13_1I 

& To find the sum of ^ and ^, both are reduced to sixths. J _ f ; 

1 — 2. 3.2 — 5 

¥ — ^ ? 6^6 — 6' 

2. When unlike parts of the same or of equal units are to 
be reduced to like parts, the number selected for the de- 
nominator should be the smallest number that is exactly 
divisible by each of the given denominators. Thus, 

To find the sum of f , f , and f , the thirds, fourths, and sixths are 
reduced to twelfths, 12 being the smallest number exactly divisible by 

3d and fi 2 — 8.3 — 9.5 — 10. 8 _i_ 9 i 10 — 27 — O 3 — Ol 

, 4, ana o. ¥ — ^ , ¥ — T % , ^ — T¥ , r^ + T2 + it — i¥ — *tj — *?* 



371. What is the value of: 



1. 


i+m? 


7. 


i+m? 


13. 


t+m? 


2. 


Kf+f? 


8. 


i+i+i? 


14. 


i+t+i? 


3. 


2 _J_ 3 i 2 ? 

3 i 3 r 3 • 


9. 


2 ill 5 V 

3 T" 6 T" 6 ' 


15. 


i+t+f ? 


4. 


__L 3 i 2 ? 
4 * 4 * 4 * 


10. 


I ill XV 
6 I 6 '1 2 • 


16. 


Hi+I? 


r>. 


2 I 4 I 6 ? 


11. 


l+f+f? 


17. 


i+i+A? 


0. 


2 14 > 4 ' 


12. 


t+A+i? 


18. 


i+t+A? 



ADDITION. 165 

372. Mixed numbers are added by finding the sums of the 
fractional and of the integral parts separately, and uniting 
the results. 

I. Required the sum of 215}, 3611 45(% and 578f. 

Process. Explanation. — /. Adding the column of frac- 

2151 tions, the sum of f , £, -J, and \ is found to be J, 

q/>-| t equal to If. The f are written under the column 

^ of fractions and the 1 carried to the units' column 

^4 of the whole numbers. 

*°T 2. The whole numbers, including the 1 from 

1605 3 the fractional column, are added in the usual 

Ij.ij.1 J.1 -1 manner and written before the fractional part, 

4 4 ^_ 1 4 4 giving 1605| for the entire amount. 

373. Rule. — I. Reduce the fractions to equivalent 
fractions having the same or a common denominator. 

II. Add the numerators and write the sum over the 
common denominator. 

III. Reduce the result, if an improper fraction, to a 
whole or mixed number. 

IV. To add mixed numbers, add the integers and 
the fractional parts separately, and unite the results. 

374. Find the sums: 

i. 261 + 131 + 241 + 321. 4 . 35f + 26f + 48f + 37|. 

*. 21i + 43i+72§ + 86§. r>. 184| + 3561 + 2741 

s. 33| + 24i + 14f + 25|. 6. 205i + 136f + 448J, 

7. 8. 9. lO. 11. 



436| 


$227f 


$4601 


1642| 


$2746^ 


527| 


$316 T V 


$755| 


2478f 


$3058| 


216^ 


$405£ 


$341| 


3654| 


$1009f 


7424, 


$213 T \ 


$152$ 


12431 


$4236f 


508f 


$7064, 


$137| 


3768f 


$39281 


207f 


$540| 


$229| 


4456f 


$17054. 



166 COMMON FRACTIONS. 

ORAL PROBLEMS. 

375. 1. Charles bought a ball for j- of a dollar, and a bat 
for ^ of a dollar. How much did both cost him ? 

2. Edith studied her history for f- of an hour and her 
geography for \ of an hour. How much time did she spend 
on both studies ? 

3. Frank caught a trout that weighed f of a pound ; John 
caught one weighing f of a pound. How much did the two 
fish together weigh ? 

Jf. Martha bought two pieces of silk ribbon ; one piece 
contained f of a yard, and the other piece, ^ of a yard. How 
much ribbon was there in the two pieces ? 

5. Three baskets of potatoes contain respectively \ of a 
bushel, i of a bushel, and ^ of a bushel. How many bushels 
of potatoes are there in the three baskets together ? 

WRITTEN PROBLEMS. 

376. i. A farmer brought to market %\ bushels of black- 
berries, If bushels of raspberries, 4| bushels of currants. How 
many bushels of berries did he take to market ? 

2. From a barrel of cider, the following quantities were 
drawn : 5f gallons, \%\ gallons, and 12^ gallons. How many 
gallons of cider were drawn from the barrel ? 

3. A man had 3 fields containing respectively 14^ acres, 
25f acres, and 15 T 9 ¥ acres. How many acres did he have in 
the three fields together ? 

4- If in one week four boys earn respectively $7^, $6f, $8|, 
and 9i, how much do they earn in all ? 

5. If from a piece of cloth, 7f yards were sold to one per- 
son, 2£ yards to another, and 19| yards remained, how many 
yards did the piece at first contain ? 

6. A grocer sold in one day the following quantities of 
sugar : 18| pounds, 24f pounds, 7| pounds, and 9^ pounds. 
How many pounds altogether did he sell ? 



1 

— ^ ., j , m 

I 



SUBTRACTION. 167 



SUBTRACTION OF FRACTIONS. 

1. If a merchant buys cloth for | of a 
dollar per yard and sells it for | of a dollar, 
does he make or lose, and how much ? 

2. Which is the greater ; f of $1, or £ 
of $1 ? f of a square inch or -J ? 

37*7. When the denominators of two 
fractions are the same, the larger frac- 
tion is the one having the larger numerator, and the differ- 
ence between the fractions is found by subtracting the smaller 
numerator from the larger, and writing the remainder over 
the denominator. Thus, 

The difference between f and f is |. f — f = f . 

378. The larger the number of parts into which a unit is 
divided, the smaller will each part be ; and when the numer- 
ators of two fractions are the same, the larger fraction is the 
one having the smaller denominator. 

1. Which is the larger, \ of an apple or \ of an apple ? 
\ of $1 or \ of $1 ? \ of $1 or \ of $1 ? \ of a unit or \ of 
the same unit ? \ or \ ? \ ox \? i or i? 

2. Which is the larger, f of $1 or f of $1 ? Which is the 
larger, f of an inch or J of an inch ? 

379. When two fractions have unequal denominators, 
before the difference of the fractions can be found they must 
be reduced to equivalent fractions having equal denominators. 
Thus, 

To find the difference between f and f , both are reduced to twelfths. 



380. Find the differences : 

1. |-i. 3. i-l. 5. f-^. 7. f-f. 

o JL J_ 4 2 2 a 2 3_ * L 4 

~- 3 4* 4 * 3 6* h ' 5 2 0' *• 2 TflT' 



1 08 COMMON FRA CTIONS. 



1. What is the value of: 










r. 


1- 

2 


-m? 



382. To subtract a mixed number from a whole num- 
ber, or from a mixed number. 

I. Eequired the difference between 2052 and 936f . 

Process. Explanation. — 1. There being no thirds 

in the minuend from which to subtract the 
f in the subtrahend, one of the units in the 



2052 



^dby minuend is reduced to thirds and the § sub- 

11 15i tracted from the §, leaving \ for the fractional 

part of the remainder. 

1 — f 5 "t — T == T 2. The integral part of the subtrahend is 

2051 — 93.6 = 1115. then subtracted from 2052, the minuend, less 

the one unit which was reduced to thirds; 

and the remainder thus obtained is written before \, the fractional 

part of the remainder, giving 1115^ for the complete remainder. 

II. Eequired the difference between 4174| and 569f. 

Process. Explanation. — i. The fractional 

l|wp parts of the minuend and subtrahend, 

£/>Q3 having different numbers for denomina- 

1— tors, are reduced to equivalent fractions 

3604^ having equal denominators, f^r^and 

t— A* Since the fractional part of the 

f 12" 4 - _ 12 minuend is the smaller, it is necessary to 

^ — l J reduce one of the units of the minuend 

l|=zf° 2% _ ^ = -11 to twelfths ; \\ + T % make ff, from 

which the -^ are subtracted, giving \\ 
for the fractional part of the remainder. 

2. The integral part of the subtrahend is then subtracted, giving 3604 
for the integral part of the remainder, which is written before the frac- 
tional part, giving 3604f| for the complete remainder. 

Note. — When the fractional part of the minuend is larger than that 
of the subtrahend, it is unnecessary to reduce to fractional parts one of 
the units of the minuend. 



SUBTRACTION. 169 

383. Rule. — I. Reduce the fractions to equivalent 
fractions having a common denominator. 

II. Subtract the numerator of the subtrahend from 
that of the minuend, and ivrite the difference over the 
common denominator. 

III. In the case of mixed numbers, treat the integral 
and the fractional parts separately. 

384. Find the remainders : 



1. 


2. 3. 


4. 


5. 


6. 


7. 


m 


36f 58^ 


42| 


m 


235-^ 


126* 


19i 


18| 20£ 


m 


_24| 


59* 


78 T V 


8. 


9. 


10. 




11. 


12. 


456f 


169 T V 


4506^ 




2540 | 


1243f 


227J 


74 f 


1754^ 




1768^ 


8561 



ORAL PROBLEMS. 

385. 1. If \ of an orange is cut from \ of an orange, how 
much will remain ? 

2. John walked \ of a mile ; Henry, f of a mile. Which 
one walked farther than the other, and how much ? 

3. Frances swam ^ of a mile ; and Katie, \ of a mile. 
Which swam farther than the other, and how much ? 

J±. Martha had a piece of ribbon containing -| of a yard and 
used f of a yard for a bow. How much ribbon remained in 
the piece ? 

5. Oliver had %\\ in his purse. What part of a dollar did 
he have left, after spending $$- ? 

6. Mr. Gray and his son together did -|f of a piece of 
work. Mr. Gray did ^ of the work ; what part did the 
son do ? 

7. Which is the greater, and how much, f of an hour or f 
of an hour ? 



170 COMMON FRACTIONS. 



WRITTEN PROBLEMS. 



386. 1. A barrel contains 31| gallons. If from a barrel 
of oil a merchant sells 9^ gallons, 15f gallons, and 6f gallons, 
how much will remain in the barrel ? 

2. In a rod there are 16^ feet. If 7 T \ feet are subtracted 
from a rod, how much will remain ? 

3. In counting the number of days' work done in two fac- 
tories, a time-keeper counted 358f days' work in one factory 
and 427f in the other. How much more work was done in 
one factory than in the other ? 

Jf. To cover one floor 35 J yards of carpet were required ; to 
cover a second floor, 28f yards. At $1 a yard, how much 
more did it cost to carpet the first floor ? 

5. Mr. Spencer had at the beginning of a month. $41f. 
He spent during the month $28 f. How much did he have 
left at the end of the month ? 

6. From a piece of cloth containing 39^ yards, a merchant 
sold 20f- yards. How many yards were left ? 

7 . A can contained 20^ gallons of milk ; after 18f gallons 
had been sold, how many gallons of milk remained ? 

8. Cloth bought at 37^ cents per yard was sold at a loss of 
8^ cents. What was the selling price ? 

9. One grocer sold 14^ pounds of sugar for one dollar ; 
another, 12| pounds. How many more pounds did the first 
grocer give for a dollar ? 

10. If in selling potatoes at I.81-J- per bushel, a merchant 
makes $.11^, how much per bushel did the potatoes cost him? 

11. A man sold 18| bushels of peaches from a wagon con- 
taining 21f bushels. How many bushels did he then have ? 

12. A lady having $35£ spent $17|. How many dollars 
did she have left ? 

13. If one man can do a piece of work in 27| days, and 
another man can do the same work in 18| days, in how much 
less time can the second man do the work ? 



MULTIPLIGA TION. 



171 



MULTIPLICATION OF FRACTIONS. 

387. Principles. — I. Multiplying the numerator of 
a fraction by any number multiplies the value of the 
fraction by that number. 



Illustration. 

3x2 — 6. — 2 
9 — 9 — 3 

Multiplying it by 
parts taken by 2. 
tion is multiplied by 2. 



Explanation. — The nu- 
merator indicates the num- 
ber of equal parts taken. 
2 multiplies the number of 
Hence, the value of the frac- 



II. Dividing the denominator of a 



fraction by any number multiplies the value of the 
fraction by that number. 

Illustration. Explanation. — If the denominator is divided by 

2 — 2_ 3 the unit will be divided into onlv one third as 

TT-s-3 — 3 

many parts, and the parts will therefore be three 
times as large. Hence, the number of parts taken remaining the same, 
their value, or the value of the fraction, is multiplied by 3. 

III. Multiplying or dividing both terms of a fraction 
by the same number does not change its value. 

Illustration. Explanation. — 1. Multiplying the nu- 

2x3 — 6. 6 — 6. merator by 3 multiplies the value of the 

fraction by 3 ; multiplying the denomina- 
tor of the result by 3 divides its value by 3. To multiply by 3 and 
divide the result by 3 evidently does not change the value. 

Illustration. 2. As before, multiplying the fraction 

- 2. by 3, and dividing the result by 3, changes 

the form, but not the value, of the fraction. 



i. 



3 — 3 ? 3 



IV. Rejecting factors common to both numerator and 
denominator does not change the value of the quotient. 

Illustrations. Explanation. — Rejecting a factor from 

8 X * X * X '3 _6_ both the numerator and the denominator of a 

>o v k x n X X. ' o K 

fraction is the same as dividing them both by 
W that factor. 



2 2 



xi 



172 COMMON FRACTIONS. 

388. To multiply a fraction by an integer or an in- 
teger by a fraction. 

I. Eequired to multiply f x 3. 

First Process. Explanation. — A fraction is 

4 x 3 _ .4x3. — i2_ multiplied by a whole number by 

t? -.3 -ijl multiplying its numerator or by 

9 9 3 dividing its denominator. By the 

Second Process. first P rocess > the numerator of the 

fraction is multiplied by 3, giving 

f X 3 == ^3 = |- = 1 ¥ for the product y - If = li. By 

the second process, the denominator 
of the fraction is divided by 3, giving for the product f = 1-J-. 

II. Required to multiply 9 by f . 

Explanation. — 9 mul- 
tiplied by f means § of 9. 
i of 9 is 3 ; f of 9 are 2 
times 3, or 6. 

Or, since the result will 
be the same, the whole number may be multiplied by the numerator and 
the product divided by the denominator. 

389. Rule. — I. Divide the integer by the denominator 
of the fraction and multiply the quotient by the numer- 
ator. Or, 

II. Multiply the integer by the numerator of the frac- 
tion and divide the product by the denominator. 

Note. — The former is the better process when the integer is exactly 
divisible by the denominator of the fraction. 

390. IVJiat is the value of: 



First Frocess. 


Second Process. 


9x| = 6 


9xf= 6 


9^-3 = 3 


9x2 = 18 


3x2 = 6 


18-j-3= 6 



1. 


Ax6? 


5. 


1 x3? 


9. 


3xA? 


13. 


W x 15 ? 


2. 


1 x2? 


6. 


i x3? 


10. 


lOx^r? 


14. 


fl x8? 


3. 


I x4? 


7. 


tVx5? 


11. 


6x&? 


15. 


_J_1_ yO? 
1 8 A * • 


4. 


1 x5? 


8. 


Ax4? 


12. 


5xl|? 


10. 


t 1 Ax12? 



MUL TIP LIG A TION. 



173. 



391. To find the product of an integer multiplied by 
a mimed number 9 or of a mioced number multiplied by 
an integer. 

I. Kequired the product of 457 multiplied by 35f . 



Process. 

457 
35f 

~342f 

2285 
1371 

163371- 

457x3 = 1371 
1371-^4 = 342f 



Explanation. — 1. The multiplicand is multi- 
plied by j, the fractional part of the multiplier, 
and the result is written as the first partial 
product. 

#. The multiplicand is multiplied by the inte- 
gral part of the multiplier, each partial product 
being written so that units of the same order stand 
in the same column. 

3. The partial products are added, giving 
16337f for the entire product. 



II. Kequired the product of 592f by 49. 



Process. 

592f 
49 

36| 
5328 
2368 
29044J 

49 x 3 = 147 
147—4 = 36} 



Explanation. — 1. The fractional part of the 
multiplicand, f , is multiplied by the multiplier, 
49 ; or, as the result is the same, the 49 is multi- 
plied by the f , and the result is written as the 
first partial product. 

2. The integral part of the multiplicand is then 
multiplied by the multiplier, 49, each partial prod- 
uct being written so that units of the same order 
stand in the same column. 

3. The partial products are added, giving 
29044f for the entire product. 



392. Rule. — I. Multiply the integer by the fractional 
part of the mixed number, and write the result for the 
first partial product. 

II. Multiply the integer by the integral part of the 
mixed number for the other partial products. 

III. Add the partial products. 



174 



COMMON FRACTIONS. 



393. Find the products: 



1. 


238 x84f. 


7. 


560 x47f. 


13. 


1099 x72^ 


0. 


4271 x 93. 


8. 


796|x37. 


14. 


2340^ x 25. 


3. 


169 x72f. 


9. 


261 x98f. 


15. 


661 x47|. 


4. 


605f x 61. 


10. 


115|x31. 


16. 


369| x 43. 


5. 


306 x36f. 


11. 


378 x49|. 


17. 


1722 xl8| 


6. 


584| x 42. 


12. 


715| x 35. 


18. 


3047| x 81. 



394. To find the product of two or more fractions or 
miaced numbers : 

I. Kequired to find the product of f and § . 



Process. 






Explanation. 
—Multiplying § 
by 2, the numer- 
ator of the mul- 
tiplier, gives £. 

But 2 is 3 times as great as the multiplier § ; 

hence, £ is 3 times as great as it should be. 

Dividing f by 3, which is done by multiplying the 

denominator by 3, gives T \. Hence, f x § =^. This result has for its 

numerator the product of the numerators of the fractions, and for its 

denominator the product of their denominators. 



II. Kequired the product of \, f , and f . 

Explanation. — The continued prod- 
uct of several fractions is only an ex- 
tension of the above process. Thus, 
the product of J x f is f ; and the 
product of f x | = ^ 6 ¥ . The same result 
is obtained by taking the continued 
product of the numerators for the nu- 
merator of the product and the con- 
tinued product of the denominators for 
the denominator of the product. 

In practice, equal factors are rejected 
from numerators and denominators before the products are found. 





First 


Process. 


1 
f 


xfxf 


— 6 — 1 

— 2T — 4 


x 


2 — 2 . 

3 — 6" 9 


2 v 3 — 
6 X 4 — g 




1X2X 


3= 6 




2X3X 


4 = 24 




A 


= i 




Second Process. 




1 x/ * 


A 4 4 



MULTIPLICATION. 175 

III. Required the product of 2f , 3f ,, f 

Process. • _ . . m , . , 

Explanation. — The mixed numbers 

^ 3- X o 4 X -§- = oj are g rs ^ reduced to improper fractions 

5 and the multiplication then performed 

^yUvl — _3_5_ — R3 , 

<£ * 4 * U — 4 — o 4 as . a bove. 



395. Rule. — I. Reduce mixed numbers to improper 
fractions and indicate the multiplication. 

II. Reject factors common to the numerators and 
denominators, 

III. Find the product of the numerators for the 
numerator of the result, and the product of the de- 
nominators for the denominator of the result, and 
reduce the result, if an improper fraction, to a whole 
or a mixed number. 

Note. — The word of written between fractions denotes multiplication. 

396. Find the products : 



1. 


fxf. 


6. 


f of | of f 


n. 


3fx5fx». 


2. 


8 X T^« 


7. 


| off off 5 . 


12. 


2| x 6| x 5|. 


3. 


fx||. 


8. 


* of 1| of f 


13. 


Ifx2§x4f. 


4. 


ttxf. 


9. 


I of | of f 


14. 


7|x.3 T Vx2|. 


5. 


Hxf 


10. 


f off of A- 

ORAL PROBLEMS 


15. 


»i x 2fV x 5|. 



397. i. If a breadth of carpet is -f of a yard wide, how 
wide will 4 breadths be ? 

2. If a boy earns f of a dollar in one day, how much will 
he earn in 8 days ? In 9 days ? 

3. How many bushels of potatoes do 10 baskets hold, if 
each basket holds f of a bushel ? 

4- There are 36 peaches on a stand and f of them are 
spoiled. How many of the peaches are spoiled ? 



176 COMMON FRACTIONS, 

5. 1 is what part of 9 ? Of 7 ? Of 5 ? Of 3 ? Of 2 ? 

6. 4 is what part of 9 ? Of 7 ? Of 5 ? Of 3 ? Of 2 ? 

7. What part of 12 is 8 ? Of 15 is 9 ? Of 11 is 7 ? 

8. What will be the cost of f of a yard of cloth at j- of a 
dollar a yard ? f of a yard at \ of a dollar a yard ? 

9. If a boy has f of an hour in which to do a piece of work 
and spends f of the time playing, how long does he play ? 
How long does he work ? 

WRITTEN PROBLEMS. 

398, 1. What will be the cost of 40 bags of sugar, each 
containing 14 pounds, at 8^ cents a pound ? 

2. A boy worked 8f hours during each of 24 days. How 
many hours altogether did he work ? 

3. How much will pay for 16 tons of coal at $5-f a ton ? 
For 8 tons at $4§ ? 

Jf. If there are 8f yards of cloth in each of 15 dresses, how 
many yards are there in all ? 

5. How much will 2| yards of cloth cost at $2 J- a yard ? 

6. If berries are worth %\ per quart, how much will 10 
boxes of 32 quarts each cost ? 

7. If blue stone for sills and lintels costs $-§• per foot, how 
much will 60 feet of it cost ? 

8. If nagging for a side-walk costs $ T 3 ¥ per square foot, 
how much will 200 square feet of it cost ? 

9. In a barrel of flour there are 196 pounds. How much 
Avdll f of it cost at 4 cents a pound ? 

10. At 16f cents, what will ^\ yards of muslin cost ? 

11. How much are 52^ pounds of sugar worth at 8^ cents 
a pound ? 

12. What will be the cost of a roll of carpet containing 
41| yards at $2£ per yard ? 

13. How much will 13 sacks of wheat cost, if each sack 
contains 2 \ bushels of wheat at %\\ per bushel ? How much 
would the wheat cost at %\ per bushel ? 



DIVISION. 



177 



DIVISION OF FRACTIONS. 

399. Principles. — I. Dividing the numerator of a 
fraction by any number divides the value of the frac- 
tion by that number. 



r 2 =i=i- 



Illustration. Explanation. — The numer- 

ator indicates the number of 
parts taken. If the numera- 
tor is divided by 2, the size of the parts will re- 
main the same, but only one half as many parts 
will be taken. Dividing the numerator by 2, 
therefore, divides the value of the fraction by 2. 



II. Multiplying the denominator of a fraction by 
any number divides the value of the fraction by that 
number. 



3. 

3X2 



— f — 



Illustration. Explanation.— The denominator indicates the 

i number of parts into which the unit is divided. 
Multiplying it by 2 multiplies by 2 the number of 
parts into which the unit is divided, and they will therefore be only one 
half as large as before. Since the number of parts taken remains the 
same, and each part is only one half as large, multiplying the denomi- 
nator by 2 divides the value of the fraction by 2. 



III. Multiplying or dividing both terms of a fraction 
by the same number does not change its value. 



2X2 
3 



i; 



4 

3"X2 



* 



Illustration. Explanation. — 1. Multiplying the nu- 

merator by 2 multiplies the value of the 
fraction by 2 ; multiplying the denomina- 
tor of the result by 2 divides its value by 2. To multiply by 2 and 
divide the result by 2 evidently does not change the value. 



Illustration. 

5-8 



- 4 . 
— 3 : 



not the value, of the fraction. 



As before, multiplying the value of 
the fraction by 2, and dividing the value 
of the result by 2, changes the form, but 



178 COMMON FRACTIONS. 

IV. Rejecting factors common to both numerator and 
denominator does not change the value of the quotient. 

Illustrations. Explanation. — Rejecting a factor from 

T ^xfx|x^ — T V both the numerator and denominator of 

i i t i a fraction is the same as dividing them 

|x|x|x^- = -gV both by that ^ctor. 

2 2 3 2 

400. Rejecting factors common to both numerator and 
denominator is called cancellation. 

401. To divide a fraction by an integer. 

I. Required to divide \ by 2. 

First Process. Explanation. — The value of a fraction 

4 _^ 2 is divided by dividing its numerator or by 

"a -7 " ~ ~I " "S multiplying its denominator. Dividing 

the numerator 4 by 2 and writing the 

quotient 2 over the denominator of the fraction gives f as the result. 

Second Process. Or, multiplying the denominator by 2 

4 . a 4 4 2 gives y 4 ^, which in its lowest terms is f , the 

5 • 5x2 To 5 same result. 

The same result is obtained by writing 
the divisor 2 in the form of a fraction, f, inverting it, and proceeding 
as in multiplication. 

402. Rule. — I. Divide the numerator or multiply 
the denominator of the fraction by the integer. Or, 

II. Write the divisor as a fraction, invert it, and 
proceed as in multiplication. 

403. To divide an integer by a fraction. 

I. Eequired to divide 2 by 4. 

Process. Explanation. — Dividing 2 by 4 gives 

\ for a quotient. But the divisor is not 
4, but |, or \ of 4. Hence, the true 
Analysis. quotient will be 5 times J, or f , equal 

2-4=1 = 4; to3 i 

t 5 9 . The same result is obtained by in- 

^ — ^ ^* verting the divisor and proceeding as in 

multiplication, 



•) _ji_ 4 — 9, v k- — U> — 91 



DIVISION. 



179 



404. Rule. — I. Multiply the integer by the denom- 
inator of the fraction and divide the product by the 
numerator. Or, 

II. Invert the divisor and proceed as in multiplica- 
tion. 



405. Find the quotients: 



> i 



5 _^Q 



2. 



4. 



k i o _j_. i 



-t-3. 8. 10-f- T V 



T6"* 



9. 
10. 



ti. 12-til. 



-if 



"2T* 



"¥2- 



*3. 


±6- 


-2. 


J4. 


2- 


_±6 
17 


15. 


1 2 _ 
17 


-5. 


16. 


7- 


. 3 
• 14 



406. To find the quotient of an integer divided by a 
mixed number, or of a mixed number divided by an 
integer. 

407. A number expressing fractional units is divided by 
another expressing like fractional units in the same manner 
as one integer is divided by another. Hence, before dividing, 
the dividend and divisor must be changed to the same frac- 
tional unit. 



I. Eequired the quotient of 136 divided by 5|. 



5-2- 

^3 



Process. 
138 
3 



17 )414(24 T 8 T 
34 

74 
68 



5* = ¥ 
138 = 4^ 



4 14 
3 



Y = 24 



TT 



Explanation. — 1. The divisor and dividend 
are both reduced to thirds. In 1 there are 3 
thirds ; multiplying the 5 by 3 and adding in 
the 2 gives 17 thirds for the divisor ; and multi- 
plying the dividend by 3 gives the number of 
thirds in the dividend. 

2. Multiply the dividend, 138, by 3 in order 
to reduce it to thirds. Having reduced both 
dividend and divisor to the same fractional 
unit, the operation is performed as in ordinary 
division. 

Note. — Both divisor and dividend are re- 
duced to the fractional unit indicated by the 
fractional part of the divisor. 



180 COMMON FRACTIONS. 

II. Eequired the quotient of 245f divided by 16. 

Process. Explanation. — 1. The divisor and dividend 

16 245 3 are k°t n reduced to fourths. 

a a %- The operations are then performed as in 

— ordinary division. 

64 ) 983 ( 15f j- Note.— 1. If the fractional part of the divi- 

64 dend had been fifths, then both divisor and divi- 

0^0 dend would have been reduced to fifths ; if 

QOn sixths, both would have been reduced to sixths, 

^T etc. 

23 2. Both divisor and dividend are reduced to 

245 3. —- 9.8.3. the fractional unit indicated by the fractional 

983 . 64 iK23 P ar ^ °f the dividend. 

— i ~ \- — 10 6~4 

408. Rule. — I. Reduce both dividend and divisor to 
the fractional parts indicated by the denominator in 
the miooed number. 

II. Divide according to the rule for dividing integers. 

409. Find the quotients : 

i. 245-5-13$. 4. 1079 -*-14f. 7. 628|~41. 

2. 178-M41 5. 832£-r-76. 8. 439f-^72. 

3. 832-T-15f. 6. 427|-~39. 9. 720 -^23|. 

410. To divide a fraction or a miooed number by a 
fraction or a mixed number. 

I. Required the quotient of f divided by f . 

Process. Explanation. — Dividing £ by 3 



4. . 3 4 x 4 i 6 — . i _i gives T \ for a quotient. But the 

5 • T — f 3— T? FF- divisor is not 3, but f, or \ of 3. 

A . . Hence, the true quotient will be 4 

' j times T \, or jf , equal to 1^. 

T~*~° — T5"5 T§' x 4 — TT — A TT* The same result is obtained by 

multiplying the dividend f by the 
divisor inverted. 



DIVISION. 



181 



II. Required the quotient of § divided by 1^. 





Process. 


Explanation. — 


2 
3 


-1.1 JL — 2_l_4 . 
• x 3 — 3 • 3 ? 


Reducing 1^ to an 


2_i_4 

3 • 3 

2-4 — 


— 2 v 3 — 6 — 

— 3 * 4 — 12 — 

2 . 2X3 — 6 


. i _ improper fraction 
t gives f . Dividing § 


3 — 


12 > 1 2 — 12 


~ 2 by | gives j% or J. 


Note. 


— When either the dividend or the divi- 


sor is a 


mixed number, 


reduce it to an improper 


fraction, 


and proceed as 


above. 



411. Rule. — I. Reduce mixed numbers to improper 
fractions. 

II. Invert the divisor and proceed as in multiplica- 
tion. 

412. Find the quotients : 



*■ m- 


s. 8i^2|. 


ft tt-5-f. 


* f-H- 


6. 4f-5-2f 


*»• *"Hf 


3. f-5-f 


7. 5f-5-3f. 


«. 41-5-151 


*■ f-H- 


*. 6f-5-4f. 


i*. 15i-4| 



ORAL PROBLEMS. 

413. 1. Robert can walk a mile in -J- of an hour. At the 
same rate how far can he walk in 4 hours ? How many fifths 
in 4 ? Divide 4 by ■$•• Divide 4 by 5. 

2. If $f is divided equally between 2 boys, how much will 
each boy receive ? $f between 2 boys ? 

c?. 3 men hired a rowboat for If per hour. How much 
did each man have to pay per hour ? 

Jf. For $f how many quarts of berries can be bought at $j 
per quart ? 

5. For $| how many yards of ribbon can be bought at &J- 
per yard ? 

6. 8 are how many times 2 ? f are how many times f ? 

7. 9 are how many times 3 ? T 9 ¥ are how many times J ? 



182 COMMON FRACTIONS. 



WRITTEN PROBLEMS. 



414. 1. If Mr. Jackson drives 7£ miles per hour, how 
long will it take him to drive 43 miles ? 

2. If Mr. Yates runs 55 miles in \\\ hours, how far does 
he run per hour ? 

3. 6 men earn equal parts of $246^ in a month. How 
much does each man earn ? 

Jf. For $5| a man purchased 6^ bushels of potatoes. How 
much did he pay per bushel ? 

5. If one man can do a piece of work in 41f hours, how 
long will it require 4 men to do the work ? 

6. How much does a farmer receive per bushel, if he sells 
365 j bushels of wheat for $408^ ? 

7. If Mr. Henry is 48^ years old and he has lived 4f times 
as long as his son, how old is his son ? 

8. If a train of cars travels 18f miles per hour, how long 
will it be in traveling 540^ miles ? 

9. If a train travels 120| miles in 4 hours, how many miles 
does it travel per hour ? 

10. A steamboat required 6^ hours to travel 92^ miles ; 
how far did it travel per hour ? 

11. What is the cost of ribbon per yard if -f of a yard costs 
5 cents ? 15 cents ? 

12. If each apron requires 2} yards of muslin, how many 
aprons can be made from 6f yards ? 

13. If 3 aprons cost $-J , how much does each apron cost ? 
ljf. If | of a yard of cloth cost $-g, how much does a yard 

cost ? 3 yards ? 3f yards ? 

15. If /f of a yard of carpet can be bought for $1|, how 
many yards can be bought for $3| ? 

16. A piece of work was performed by 18 men in 25^ days. 
In what time could 38 men have done the work ? 

17. If 8 equal pipes can discharge a cistern in 23^ hours, 
how long will 3 of the pipes be in discharging it ? 



DECIMAL FRACTIONS. 



183 



DECIMALS. 




.01 



9 
.001 



415. Decimal parts of units are fractional units ob- 
tained by dividing an integral unit into tenths, hundredths, 
thousandths, etc. 

The word decimal is derived from the Latin word decern, which sig- 
nifies ten. 

416. A fraction composed of one or more of the decimal 
parts of a unit is called a decimal fraction, or a decimal. 

417. For convenience, only the numerator of a decimal 
fraction is written, and the denominator is indicated by 
placing a period, or a period and one or more ciphers, before 
the numerator. Thus, 

One tenth 

Three tenths 

One hundredth 

Twenty-five hundredths 

Two hundred twenty-five thousandths 

Sixty-five ten-thousandths 



10 
3^ 

10 

1 

100 

25 
100 
225 
1000 

65 
10000 



Orders. 



Names of 
Places. 



Number. 



Integral Orders, 



6th. 5th. 4th. 3d. 2d. 1st. 



is written 


.1 


a it 


.3 


a a 


.01 


a a 


.25 


" 


.225 


a a 


.0065 



Deeimal Orders. 

, * 1 

1st. 2d. 3d. 4th. 5th. 6th. 



• 03 





I 
S3 


o3 

CO 
O 




co 

S3. 

o3 
co 
S3 
O 


CO 

<x> 
u 
-a 
S3 

3 


co 


CO 

"3 


a 


H 


H 


w 


H 


£ 



4* 
•iH 

o 









t3 
s3 

S3 
W 



PI 

o3 

CO 

O 



4 6 9 8 5 



6 4 3 



§ 

CO 

■S3 
O 
^3 







S3 

o3 

CO 

S3 
O 



S3 



5 



184 



DECIMAL FRACTIONS. 



418. The period is called the decimal point ; places to 
the left of the decimal point are called integral places or 
orders ; places to the right of the decimal point are called 
decimal places or orders ; and ciphers between the deci- 
mal point and the first significant decimal figure are called 
decimal ciphers. 

1. The first decimal place expresses what ? The second ? 
The third ? The fourth ? 

2. How many hundreds in 1 thousand ? How many tens 
in 1 hundred ? How many units in 1 ten ? How many 
tenths in 1 unit ? How many hundredths in 1 tenth ? How 
many thousandths in 1 hundredth ? 

419. The number of decimal places is always equal to the 
number of ciphers which would be in the denominator, if the 
decimal were written as a common fraction. 

420. Express decimally the following fractions : 



A- 

TOO"' 
5 


4. 
5. 
6. 


5 5 
100* 

15 _ 

1 oTfo* 

5 


7. 
8. 
9. 


_1 5 5 
10 0* 

1 55 
10 00 

7 05 


100 0* 


10 0* 


1000 



421. Since the different orders of decimals decrease from 
left to right in the same ratio as in the different orders of 
integral units, a decimal and an integer may be written 
together in a single number. Thus, 



7 and ^ are written 7.7 
7 and yfo " " 7.07 

27 and T %\ " " 27.27 



27 and T fJ 7 are written 27.027 
27 and jffifc " " 27.0127 



27 and T oWi>o 



27.00127 



7000, 700, 70, 7, y^, T fo, T oW ma y a U ^ e expressed as a single num- 
ber; thus, 7777.777. 

422. Numbers composed of integers and decimal fractions 
are called decimals, or mixed decimals. 



DECIMAL FRACTIONS. 185 

423. To read a decimal, read the number expressed by 
the figures at the right of the decimal ciphers, and then give 
the name of the last decimal figure. 

Thus, .025 is read 25 thousandths ; .0017 is read 17 ten-thousandths. 

424. In reading a number composed of an integer and a 
decimal, the decimal point is read and. 

Thus, 5234.2027 is read: five thousand, two hundred thirty-four, and 
two thousand, twenty-seven ten-thousandths. 

425. Write from dictation and read: 



1. 


.7 


6. 


4.04 


n. 


3.3 


16. 


5.0005 


ft. 


.005 


7. 


.0001 


12. 


4.09 


17. 


600.06 


3. 


.09 


S. 


19.19 


13. 


7.17 


18. 


4.0404 


4. 


.08 


9. 


7.078 


14. 


10.1 


19. 


3.3003 


5. 


7.07 


10. 


30.003 


15. 


17.01 


20. 


900.09 



426. Express decimally and read: 

o 3 4 11 fi 7 15 o 1001— J 

^•10 00 *• 100 000 "• M000 9 ' iW1 10 00 

427. Express in figures as decimals: 

1. Ten, and one tenth. 

2. Forty-four hundred, and forty-four hundredths. 

3. Three, and three ten-thousandths. 

J/,* One hundred thousand, and one hundred-thousandth. 

5. One hundred one, and one hundred one ten-thousandths. 

6. Seven thousand, seventy, and seven hundredths. 

7. Eighty, and eighty-one thousandths. 

8. Ninety-one thousand, ninety-one, and ninety-one thou- 

sandths. 

9. One hundred thousand, ten, and eleven ten-thousandths. 



186 DECIMAL FRACTIONS. 



PRINCIPLES OF DECIMALS. 

428. I. Annexing or rejecting a cipher at the right 
of a decimal does not affect its value. 

Illustration. Explanation. — Writing .3 

3 — _3_ • J_ y l o — .3_o_ — 30 as a common fraction and mul- 

•° — 10? 10 A io — 100 — ' OKJ 

tiplying both terms by 10 gives 
y 3 ^, or .30. Conversely, writing .30 as a common fraction and dividing 
both terms by 10 gives .3. 

II. Inserting a, cipher between a decimal and the 
point divides the decimal by 10. 

Explanation. — Writing .3 

Illustration. as a common fraction and 

.3 = ^-; -^-^-10 = yI-q = .03 dividing the fraction by 10 

gives yfo, or .03. 

III. Rejecting a cipher from the left of a decimal 

multiplies the decimal by 10. 

Explanation. — Writing 
Illustration. .03 as a common fraction 

.03 = t |"q ; yf-g- X 10 = T 3 ¥ = .3 and multiplying the frac- 

tion by 10 gives y 3 ^, or .3. 

429. To reduce a common fraction to a decimal : 

I. Required to reduce £ to a decimal. 

Process. Explanation. — Ciphers are 



a \ i qa annexed to the numerator 1, 

^r the resulting number divided 

by the denominator 4, and as 
Analysis. many decimal places pointed off 

1 100 in the quotient as there were 

J- = V 1 — -f 11 — iVo = '25 ciphers annexed. 

In the analysis, the numera- 
tor 1 is both multiplied and divided by 10, giving T g, a result which, if 
divided by the denominator 4, gives .2J, a complex decimal. 

Multiplying both terms of T $ by 10 gives T g$ ; dividing T $# by the 
denominator 4 gives T 2 ^, or .25, a simple decimal. 

430. Rule. — I. Annex ciphers to the numerator and 
divide the resulting number by the denominator. 



REDUCTION. 187 

II. Point off in the quotient as many decimal places 
as there were ciphers annexed, prefixing decimal 
ciphers, if necessary. 

Note. — If the denominator of a fraction is neither 2 nor 5 nor some 
product of these numbers only* the fraction cannot be reduced to a sim- 
ple decimal. In such cases, the fraction may be changed to a complex 
decimal, or the division may be carried to several places and the re- 
mainder disregarded. In this case, the sign + is used to show that the 
value is approximate. 

Thus, f = .28|, or .2857+ ; i = .8J= .333 + . 

431. Reduce the following to equivalent decimals: 

-/111 olll /r_3r>f4 v 15 JUL JLL 

o 2. 3 _5 4 1 2. 5. * 1. f 1 f 1 c 43 1_J_3 _1_ 

^' 4> T> 8' ** ^> 3> 6* °* 2 UL 8 U1 3* *" 8 0> 1 2 8> 3 2* 

432. To reduce a decimal fraction to a common 
fraction. 

I. Required to reduce .75 to a common fraction in its 
simplest form. 

Process. Explanation. — The decimal .75 is written as 

wk 75 a common fraction j 5 ^. Both terms of y 7 ^ are 

" g iVA 3 divided by any convenient common factor, as 5, 

ioo Yo ¥ and both terms of the resulting fraction are 
divided in like manner. The terms of £ con- 
taining no common factor, the fraction is in its simplest form. 

433. Rule. — Express the decimal as a common frac- 
tion and reduce it to its simplest form. 

434. Change the folloiving decimals to equivalent frac- 
tions in their simplest form : 

i. $.08. 3. 175. 5. .004. 7. .005. ». .308. u. 1.625. 
2. $.25. 4. .035. e. .045. s. .025. io. $.125. *#. $.875. 

Note.— Complex decimals and mixed decimals arc reduced in the 
same manner as simple decimals. 

1J.2 IOO 191 2 5 

TVino 1/12 — 1H T__~T~ l. -(91 x ^2 _ ~g~ — 25 — 5 — 1 

Thus, .14^-— -— - 7 , .12.-— -^-^-^-^ 



188 DECIMAL FRACTIONS. 

ADDITION AND SUBTRACTION OF DECIMALS. 

435. To find the sum of two or more decimals. 

I. Required to find the sum of 13.25, 150.5, and 4.06. 

Process. 

13.25 Explanation. — The decimals are written so that 

150 5 ^oth integral and decimal units of the same order 

a r\r* stand in the same column, and the addition is then 

— performed as in whole numbers. 

367.81 

436. To subtract one decimal from another. 

II. Required to subtract 16.007 from 42.05. 

Process. Explanation. — The decimals are written so that 

42.05 both integral and decimal units of the same order 

16.007 stand in the same column, and the subtraction is then 

9fi a^o performed as in whole numbers. 

Note. — A vacant decimal order, as thousandths in the minuend 
above, is treated as if filled with a cipher, since .01 =.010. 

437. Rule. — Write the numbers so that units of the 
same order stand in the same column, and then per- 
form the addition or subtraction as in whole numbers. 

438. Write from dictation and find the sums : 



1. 


2. 


3. 


4. 


500.05 


70.0071 


4.0004 


200.2 


68.006 


100.01 


37.0307 


2.02 


306.1 


45.045 


50.5 


79.079 


45.067 


3.06 


5.053 


1.1 


5. 


c. 


7. 


s. 


.01802 


1.011 


404.028 


.101 


.0603 


16.002 


3.009 


10.01 


.0012 


.0707 


.507 


1.0001 


.202 


.5005 


80.08 


.01 



SUBTRACTION. 189 

439. Write from dictation and find the differences : 

1. 2. 3. 4. 

$14,086 $17,000 50.005 $400.04 

7.098 8.095 9.09 37.07 



5. 


6. 


7. 


8. 


20.02 


900.09 


.0047 


1.0101 


2.20 


88.88 


.0009 


.101 



WRITTEN PROBLEMS. 

440. i. A piece of cloth was bought for $| per yard and 
sold for $.45 per yard. How much per yard was the gain 
or loss ? 

2. If muslin is bought for %\ per yard and sold at a profit 
of $.02|, what is the selling price ? 

3. Find the smaller of two numbers,, the greater of the 
two numbers being 500.5 and the difference 75|. 

^. The difference between two numbers is 180.08 ; the less 
number is 750.125. Find the other number. 

5. If the sum of three numbers is 100.001,, and two of the 
numbers are 50.05, and 37.3, what is the third number ? 

6. From a rod subtract 5.25 feet. What is the remainder ? 

7. What will be the entire cost of a yard of each of 5 dif- 
ferent qualities of silk, costing respectively : $.75, $1£, $lf, 
$1.95, and $2|? 

8. From a barrel containing 2|- bushels of apples 1.375 
bushels were taken. How many bushels were left ? 

9. If a lady pays $.60 for a bottle of cologne, $| for each 
of 2 brushes, and $ \ for each of 3 bottles of different extracts. 
How much in all does she pay ? 

10. Mamie bought 3 bunches of violets at $.05 each, 
10 roses at $.10 each, a bunch of geraniums for $.08, and 
7 pinks at $.03 each. How much change should she receive 
out of a $2 bill ? 



190 DECIMAL FRACTIONS. 

MULTIPLICATION AND DIVISION OF DECIMALS. 

441. To multiply or divide by 10, 100, 1000, etc. 

I. When the multiplicand consists of an integer and a 
decimal written together, the product is found by moving the 
decimal point to the right as many places as there are ciphers 
in the multiplier, annexing ciphers when necessary. 

Thus, to multiply 178.25 by 1000, annex one cipher, move the decimal 
point three places to the right, and the result, 178250, is the product. 

II. When the dividend is an integer, the quotient is found 
by pointing off from the right of the dividend as many places 
as there are ciphers in the divisor. 

Thus, to divide 1887 by 100, point off two places, and the result, 
18.87, is the quotient. 

Process. Explanation. — Dividing in the regular way 

100 ) 1887 ( 18 87 gives a quotient of 18 and a remainder of 87, 

-I nr which, when written over the divisor and an- 

. nexed to the quotient, gives for the entire quo- 

887 tient 18^, or 18.87. 

800 The number of places pointed off in the quo- 

~~ZZ tient, 18.87, is the same as the number of ciphers 

in the divisor. This will evidently be true what- 
ever may be the number of ciphers in the divisor. 

III. When the dividend is an integer and a decimal written 
together, the quotient is found by moving the decimal point 
as many places to the left as there are ciphers in the divisor. 

Thus, to divide 2375.19 by 100, the decimal point is moved two places 
to the left, giving 23.7519 for the quotient. 

442. Find the value of: 

1. 641x100. 5. 862-r-lO. o. 465.5-t-IOO. 

2. 32.2x100. 6. 973 — 100. 10. 465.5x100. 
s. 243.2x1000. 7. 1560-M00. u. 30.07x1000. 
*. 1.257x100. s. 8700-H000. **, 30.07-r-lOOO, 



MULTIPLICATION AND DIVISION 191 

443. To find the product of two or more decimals, 

I. Kequired to multiply .025 by .07. 

Process. Explanation. — The decimals are so 

Q25 written that the multiplication may be per- 

^ formed the most readily. 25 is multiplied 

— by 7 as if both were integers, and decimal 

.00175 ciphers are prefixed to 175 until the prod- 

uct contains as many decimal places as the 
na ysis. multiplicand and multiplier together. 

.025 = xoo~o The analysis shows why the product is 

Qiy __ _7 thus pointed off. The decimals are written 

as common fractions, and the multiplica- 
Toiro" x TTTo" — iooo'oo tion is performed. The number of ciphers 

in the denominator of the product is equal 
to the number of ciphers in the denominators of the multiplicand and 
the multiplier together. It follows, therefore, that .025 x .07 =.001 75. 

444. Rule. — Perform the multiplication as in in- 
tegers, and point off as many decimal places in the 
product as there are in the multiplicand and the mul- 
tiplier together. 

445. Find the products : 



1. 


107.5 x. 3. 


■ 5. 1.03 x. 65. 


9. 


700.07x1.3. 


2. 


29.06 x. 5. 


o. 20.5x4.07. 


10. 


.09 x. 0001. 


3. 


100.7 x. 23. 


7. .0036x2.1. 


11. 


1.001x150. 


4. 


40.4x3.3. 


s. .505x1.05. 


12. 


35.03x300.5. 



446. To find the quotient of one decimal divided by 
another. 

I. Required to divide 1.5 by .04. 

Process. Explanation. — One cipher is annexed to 1.5, so 

.04 ) 1.50 o that the number of decimal places in the dividend 

niZT may be equal to the number of decimal places in the 

divisor. If 1.50 were exactly divisible by .04, the 
quotient would be a whole number, since an integer multiplied by .04 
would give a product with two decimal places. There being a remain- 
der, however, additional ciphers are annexed to the dividend until the 
division is exact, or until the result is sufficiently accurate for practical 
purposes. 







1.31 






Analysis. 




.524+ 


.4: 


- 524 jl 

— To o o • 


. 4 — 
TO — 


l 3 1 












— 13 1 — 

— 100 — 


:1.31 


1 









192 DECIMAL FRACTIONS. 

II. Required to divide .524 by .4. 

Process. Explanation. — The division is per- 

.4) .524 formed as in whole numbers, and as 

many decimal places pointed off in the 
quotient as the decimal places in the 
dividend exceed in number those of 
the divisor. 

The analysis shows the reason for 
thus pointing off the quotient. By the 
inversion of the divisor each cipher it 
contains cancels a cipher in the de- 
nominator of the dividend. The denominator of the product, therefore, 
contains as many ciphers less than the denominator of the dividend as 
there are ciphers in the denominator of the divisor. Hence, it follows 
that .524 -j- .4 = 1.31. 

III. Required to divide .0064 by .4. 

Process. Explanation. — The division is 

4) 0064 performed as in whole numbers. 

Decimal ciphers are prefixed to the 

•"•1" significant figure of the quotient 

Analvsis until the quotient contains as many 

decimal places as the number of 

.UUo4: — .4 to~ooo"~ ; ~To == decimal places in the dividend ex- 

*f „« ., „ ^ „ ceeds the number of decimal places 

i o o o¥ * 4 — i o o o — - uxu i n the divisor. 

447. Rule. — Annex ciphers to the dividend, if neces- 
sary ; divide as in whole numbers, and point off the quo- 
tient so that the decimal places of the divisor and quo- 
tient shall together be equal to the decimal places of the 
dividend. 

448. Find the quotients: 

i. 31.5-h.126. 4. 1. 69 + . 013. 

2. 1.718 + .12. r>. 884.-f-.026. 

3. 5.2 + . 005. o. .0625 + . 05. 

449. Find quotients to three decimal places : 

i. 23.5 + 9.3. 3. 20.2 + .006. 

2. $123 + 609, *. $50,05-K.13. 



7. .66-5- 


.015. 


s. 7.2-5- 


-360. 


o. 157-5- 


-.013. 


ilaces : 




s. 100- 


5-. 33. 


e, 25- 


r-6.06, 



MULTIPLICATION AND DIVISION 193 

WRITTEN PROBLEMS. 

450. 1. A man having $116.50 spent $89.35. How much 
did he have left ? 

2. A surveyors chain containing 100 links of 7.92 inches 
each. What is its length in inches ? In feet ? 

3. In a bushel there are 2150.42 cubic inches. How many 
cubic inches are there in a quart ? 

Jf. In 66 feet how many rods are there, each rod being 16.5 
feet long ? 

5. Mr. Peck owns .125 of a farm worth $1625. What is 
his share worth ? , 

6. A company bought potatoes for $2680.50 at $.80 per 
bushel. How many bushels did the company buy ? 

7. How many yards are there in a piece of cloth worth 
$32.25 at $.50 per yard ? 

8. At $.32 a yard, what will 106.5 yards of chintz cost ? 

9. How much are 52^ pounds of sugar worth at 8^ cents a 
pound ? 

10. If each box holds 8£ pounds of tea, how many boxes 
will be required to hold 152.5 pounds of tea ? 

11. If each barrel contains 31.5 gallons, how many barrels 
will be required for 12600 gallons of oil ? 

12. How much will \ of 1000 bushels of apples cost at $.25 
per bushel ? 

13. If .75 of an acre cost $375, what will 16 acres cost ? 
i^. A merchant owning .375 of a ship worth $125,000, sold 

his share for $47,630. Did he make or lose, and how much ? 

15. If Mr. Drake can do f of a piece of work in 10 \ hours, 
how long will he require for \ of the work ? 

16. Mr. Barlow paid $742.50 to 30 men for 8i days' work. 
How much per day was this for each man ? 

17. In how many days will 50 men earn $625, if each 
man receives $2.50 per day ? 

18. How much per day does each of 40 men earn, if by 
working 25 days they together earn $2 ; 500 ? 



SECTION VI. 



UNITED STATES MONEY. 

451. United States Money is a decimal currency. The 
unit is 1 dollar. Cents are expressed as hundredths of 1 dol- 
lar and mills as thousandths. 

1. How many cents United States Money. 

are there in 1 dollar ? 10 mills make 1 cent, $, or ct. 

2. How many mills 10 cents " l dime, d. 
are there in 1 cent ? In 10 dimes " 1 dollar, $. 
1 dime ? In 1 dollar ? 10 dollars " l eagle, E. 

3. How many dimes 

are there in 1 dollar ? In 1 eagle ? 

Jf. 1 cent is what part of 1 dime ? Of 1 dollar ? 

5. 1 dime is what part of 1 dollar ? 

6\ 1 mill is what part of 1 cent ? Of 1 dime ? Of 1 dollar ? 

7. 1 dollar is what part of 1 eagle ? 

452. The decimal point is placed between dollars and 
cents. Since there are 100 cents in 1 dollar, two decimal 
places are given to cents, and the third decimal place, or 
thousandths, is given to mills. 

Mills may be regarded as tenths of cents, dimes as tenths 
of dollars, and eagles as tens of dollars. 

Thus, forty-five dollars, seventy-five cents, three mills, is written 

$45,753. 



UNITED STATES MONET. 195 

453. Express in figures the following amounts : 

1. Four dollars, fifty five cents. 

2. Five dollars, five cents. 

• 3. One hundred five dollars, one cent, five mills. 
Jj,. Three hundred dollars, seven cents, five mills. 

5. Forty-seven dollars, forty-seven mills. 

6. Sixty dollars, sixty cents, six mills. 

7. One thousand dollars, one mill. 

454. To reduce cents and mills to dollars^ or dollars 
to cents and mills. 

455. I. Cents are reduced to dollars by dividing by 100, or 
by pointing off two places ; and mills are reduced to dollars by 
dividing by 1000, or by pointing off three places at the right. 

Thus, 2045 cents = $20.45, and 2045 mills = $2,045. 

II. Dollars are reduced to cents by multiplying by 100, 
or by annexing two ciphers ; dollars are reduced to mills by 
multiplying by 1000, or by annexing three ciphers ; dollars 
and cents are reduced to mills by annexing one cipher and 
dropping the decimal point. 

Thus, $25 = 2500 cents, or 25000 mills; $14.04= 14040 mills. 

456. Express as dollars and cents : 



1. 


2750 cts. 


4. 40400 cts. 


7. 


40025 m. 


£. 


4030 cts. 


5. 125000 m. 


8. 


100001| cts, 


3. 

3? 


89065 m. 
'. Express as 


o. 27535^ cts. 
cents 9 or as mills : 


9. 


600600 cts. 


l. 


$38. 


4. I10.16J-. 


7. 


$4000§. 


2, 


$50.50. 


s. $100. 03f 


S. 


$6060f 


3, 


$42.37f 


o. $200£. 


9. 


$7700f. 



196 UNITED STATES MONET. 

458. To add or subtract in United States Money. 

459. Numbers expressing United States Money are added 
or subtracted the same as decimals. 

460. Write from dictation and add : 



1. 


2. 


3. 


4. 


$40.72 


$507.75 


$1900. 


$110,105 


404.31 


69.90 


2010.30 


63.063 


63.49 


703.02 


463.06 


770.707 


728.03 


50.05 


1009.80 


80.08 


49.90 


500.50 


550.50 


909.909 



461. Write from dictation and find the differences : 

1. 2. 3. 4. 

$100,705 $505.05 $458,008 275.07 

46.069 370.76 399.099 88.075 



462. To multiply in United States Money. 

I. Required to multiply $25.12 by 34. 

Process. 

).12 Explanation. — The multiplicand is considered as 

34 expressing 2512 cents. The multiplication is per- 

formed as in whole numbers, and two places are 
pointed off from the product to reduce it to dollars. 



10048 
7536 

$854.08 



II. Required to multiply $16.37^ by 42. 

Process. Explanation. — The multiplicand may be consid- 

$16,375 ered as expressing 16375 mills, and multiplied as in 

42 whole numbers. Three places are then pointed off to 

32750 reduce the product to dollars. Or, 

65500 The same result will be obtained if the multipli- 

cand is considered as expressing 16374 cents, the mul- 
tiplication performed as in mixed numbers, and two 



$687,750 
places pointed off from the product to reduce it to dollars 



UNITED STATES MONEY. 197 

463. Find the value of: 

i. $40.45 x 38. 5. $27.42| x 290. 9. $18.32 x 23|. 

2. $29.09x97. e. $30.03^x44. 10. $25.05 x81|. 

3. $60.62x50. 7. $566,075x32. n. $140.73 x 69|. 

4. $450.35x93. 8. $2345.75^x33. 12. $205.00 x 55f 

464. To divide in United States Money. 

I. Required the quotient of $12.48 divided by 25 cts. 

Process. Explanation. — The dividend is con- 

$.25 ) $12.48 (49 If sidered as expressing 1248 cents and the 

100 division is performed as in whole num- 

24g bers. The dividend and divisor are both 

225 concrete, and the quotient is therefore 

"00 abstract. 

Note. — When the dividend and divisor 
are both concrete, before dividing they must be made to express the 
same denomination, dollars, cents, or mills. 

II. Required the quotient of $25,725 divided by 75. 

Process. Explanation. — The dividend is con- 

75 ) $25. 725 ( $.343 sidered as expressing 25725 mills, and the 

22 5 division is performed as in whole num- 

3 22 bers. The divisor is abstract, hence the 

3 00 quotient is like the dividend and expresses 

— jJqk 343 mills. To reduce it to dollars three 

905 places are pointed off. 

Note. — When the dividend expresses 
simply dollars and does not contain the divisor an exact number of 
times, reduce the dividend to cents before dividing. 

465. Find the quotients: 

1. $625~$25. s. $306.40-^$44. 

2. $4.32-^39. e>. $75,075—25. 

3. $550-r-$51. 7. $8.808-f-504. 

4. $660.60-r-$32.20. s. $27~$3.65. 



9. 


$130.25-, 


1-25. 


10. 


$200.00- 


1-50. 


11. 


$325.75^ 


3-12*. 


IS. 


$490. 50 -. 


r37|. 



198 UNITED STATES MONET. 



FRACTIONAL PARTS OF ONE DOLLAR. 

466. Operations in multiplication and division are fre- 
quently shortened by considering the cents as fractional parts 
of a dollar. 

467. Copy and commit to memory : 



50 =$f 


$.25 = $i. 


$.75 =$}. 


33| = H- 


$.16$ =:$!-. 


$.66f = $f. 


20 = *f 


$.10 =$^0- 


$.05 = $ 5 i 


12i = $£. 


$.374- = If. 


$.62|- = $f. 



ORAL PROBLEMS. 

468. 1. A grocer sells butter for 25 cents or $J per pound. 
How much does he receive for 20 pounds ? 

2. If ribbon is worth 33| cents a yard, how much will 12 
yards cost ? 

3. What will 33 J yards of ribbon cost at 12 cents a yard ? 
^. At 20 cents each, what will 25 water-melons cost ? 

5. When calico is sold for 12 \ cents a yard, what will 16 
yards cost ? 

6. A laborer receives 16f cents an hour for his work. How 
much does he receive for 12 hours' work ? 

7. What will 15 hats cost at 66f cents each ? 

8. If straw hats cost 50 cents each, how many can you 
buy for $8 ? 

9. How many halves are there in 1 ? In 8 ? In 15 ? 

10. At 33| cents per yard, what will 30 yards of lace 
cost ? 

11. How many thirds are there in 1 ? In 5 ? In 30 ? 

12. For how many bushels of apples at 37| cents will $3 
pay? 

13. How many eighths are there in 3 ? - 2 8 4 -r"! = ? 



UNITED STATES MONET. 199 

ljj,. At 62 § cents a bushel, how many bushels of pears can 
you buy for $4 ? 

15. In 4 how many eighths are there ? - 3 -g 2 --f-f — ? 

16. $5 will buy how many pairs of gloves at 66| cents a 
pair ? 

WRITTEN PROBLEMS. 

469. 1. What will be the cost of 468 bushels of wheat at 
$1.33| a bushel ? 

^. Find the cost of 880 bushels of rye at $.625 a bushel. 

3. How much will 66 tons of coal cost at $5.33^ a ton ? 

Jj,. At 75 cents a yard, how many yards of cassimere can 
be bought for $39 ? 

5. If it costs $36 to carpet a room with carpet at $1.12-§- 
a yard, how many yards of carpet are required ? 

6. What will 720 men earn in 1 hour at $1 per hour ? At 
50 cents ? At 25 cents ? At 33-J- cents ? 

7. For how many linen collars at 25 cents each will $25 
pay? 

8. $37 will pay for how many weeks' board at $4.62| a 
week ? 

9. Find the entire cost of 18 pieces of muslin, containing 
36 yards each, at $.16f a yard. 

10. How many pounds of tea can be bought for $25, if 
each pound costs $. 62^ ? 

11. How many collars can be bought for $40, if each collar 
costs $.12|- ? 

12. If a dozen buttons are worth $.37^, what will 10-f 
dozens be worth ? 

13. How much will be made on 18f yards of calico, cost- 
ing 10 cents a yard, and sold for 12| cents a yard ? 

lJj,. How many pounds of sugar at 5 cents a pound will be 
worth as much as 60 pounds of coffee at 33^ cents a pound ? 

15. How many yards of cloth can be bought for $35, if 
one yard costs $.87^ ( = $1) ? 



200 



UNITED STATES MONEY, 



BUSINESS FORMS. 

470. A debtor is one who owes money, goods, or service. 

471. A creditor is one having a claim against a debtor 
for money, goods, or service. 

472. A bill is a formal statement of debits and credits, 
with dates, place of business, and names of parties. 

When the creditor, or his agent, writes or stamps on a bill the ex- 
pression Received payment, or Paid, and signs his name to the bill, it is 
said to be receipted. 



473. 

Mr. Philip S. Bryce, 



Augusta, Me., Sept. 6, 1886. 



Bought of Henry McLean & Co., 



25 lb. 

18 " 

H " 

10 « 

86 " 

5 " 



Coffee . 
Tea . 
Rice . 
Candles 
Sugar . 
Starch 



$.25 
.40 
.05 
.15 
.07 
.10 




Received payment, 

Henry McLean & Co., 
Per C. L. 



474, When, according to the bill, were the items men- 
tioned above purchased ? At what place ? By whom ? Of 
whom ? By whom was the bill made out ? Who paid the 
bill ? By whom was it receipted ? What does per C. L. 
mean ? What is the amount of the extension of the first 
item ? Of the second ? Of the third ? Of the fourth ? Of 
the fifth ? Of the sixth ? What is the footing of the bill ? 



BUSINESS FORMS. 



201 



475. The following abbreviations are used in bills : 



@, 


At. 


Cr., 


Creditor. 


Net, Without disc't 


Acc't, ; 


% Account. 


Dr., 


Debtor. 


No.orti, Number. 


Am't, 


Amount. 


Do. or Ditto, 


The same. 


Pay't, Payment. 


BaL, 


Balance. 


Doz., 


Dozen. 


Pd., Paid. 


Bbl., 


Barrel. 


Inst., 


This month. 


Per, By. 


Bo't, 


Bought. 


Int., 


Interest. 


Rec'd, Received. 


Co., 


Company. 


Mdse., 


Merchandise. 


Ult., Last month. 



476. Copy and find the footings of the following bills . 

(i.) St. Louis, May 1, 1885. 

Mr. Robert Crane, 

Bought of John Halifax & Co., 



5 lb. Bon Bons . . 

3 " Mottoes, . . ... 
6. : " Caramels . . 

4 " Burnt Almonds 



$.75 
.25 
.60 
.50 



Received payment, 

John Halifax & Co. 



{2.) New York, Aug. 14, 1886. 

Mrs. William Armstrong, 

To Arnold, Billings & Co., Dr. 



1886. 


Jan. 


15 


a 


a 


ii 


21 


Feb. 


s 


it 


7 


ii 


a 


Mcli. 


15 


ii 


a 



To 25 yds. Silk 



Jfi 



15 

H 

6 
S 



Sheeting . 
Flannel . 
Gingham . 
Silk Flush 



@ $1.75 
@ .20 
@ .60 
@ .18 
@ 15.00 



Quilted Satin @ 1.25 
Velvet . . @ 2.15 
2 doz. Handkerchiefs®, 8.00 



Received payment, 

Arnold, Billings & Co, 



202 UNITED STATES MONEY, 

4HH. 1. Promissory Note : 
$500. New York, Aug. 7, 1886. 

. . . Three months after date .../.. .promise to pay 

to the order of Samuel Simpson, 

Five Hundred Dollars, 

at the National City Bank. 

Value received. Due Nov. 7/10. Richard Roe. 

2. Bank Check: 

No. 791. Brooklyn, Sept. 8, 1887. 

National City Bank, 
of Brooklyn. 

Pay to the order of Matthew Morrison 

One Thousand Dollars. 

$1000^%. Frederick Styles. 

3. Due Bill : 

Baltimore, Jan. 16, 1875. 

Due, L. W. Smith, 

One Hundred Twenty-five Dollars, 

payable in goods from our store. 

Emerson, Dudley & Co. 

4- Receipt on Account: 

$650. San Francisco, Dec. 9, 1885. 

Received from Buell, Bolton & Co., 

.... Six Hundred Fifty Dollars 

on %. Robert Marlow. 

5. Receipt in Full of Account : 

S560/jf w . Cincinnati, July 4, 1886. 

Received from Arnold, Archibald & Co., 

— Five Hundred Sixty t ^ • • Dollars, 

in full of all demands to date. 

Paul Pry. 



INTEREST. 203 



INTEREST. 

1. If 1 hundredth or 1 per cent, of $1 is $.01, how much 
is .02 or 2 per cent. ? How much is .03 or 3 per cent. ? 

2. If 4 per cent. (4 %) is paid for the use of $1 for 1 year, 
how much is the interest or the sum paid for the use of the 
money ? 

3. If the interest of a sum of money is b% for 1 year, how 
much will it be for 2 years ? For 3 years ? For 4 years ? 

4. How much is 1% of $1 ? Of $2 ? Of $3 ? Of $7 ? 

5. How much is 1% of $100 ? Of $200 ? Of $300 ? Of 
$700? 

6. Find 2% of $100 ? Of $200. Of $400. Of $500. If 
$10 is 2% of $500, how much is ±% ? 

7. What is the interest on $100 at 5% for 1 year ? For 2 
years ? For 3 years ? For 5 years ? 

8. What is the interest on $100 at 5% for 1 year ? On 
$200 ? On $500 ? On $900 ? 

9. What is the interest on $200 at 1% for 1 year ? For 2 
years ? For 3 years ? 

10. What is the interest on $200 at 2% for 1 year ? For 2 
years ? For 5 years ? 

11. What is the interest on $300 at 2% for 5 years ? On 
? On $1000 ? 



478. Interest is money paid for the use of money. 

479. The principal is the money for the use of which the 
interest is paid. 

480. The rate of interest is the number of hundredths of 
the principal paid for interest in one year. 

481. The amount is the sum of the principal and the 
interest, 



204 INTEREST. 

482* To find the interest on any sum of money at any 
rate for years and months. 

I. Find the interest on $235 at 1% for 3 yrs. 6 mos. 

Process. Explanation. — The interest 

$235 = Principal. for 1 ^ ear is 7 # or '° 7 of the 

~ry r> ±. principal. 3 years 6 months 

equals 3J years. The interest 



$16.45 = Int. 1 yr. 

3|- = Time. terest for 1 year. 

qoojl Note. — The interest for 1 mo. 

aqq~ 2 is y 1 ^ of the interest for 1 year; 

and the interest for 1 day is 3^ 

$57.57| = Int. 3 yrs. 6 mos. of the interest for 1 month. The 

interest for years, months, and 
days may be found separately and then added, the result being the in- 
terest for the entire time. 

483. Rule. — Multiply the principal by the per cent, 
expressed in hundredths and that result by the number 
expressed in years, 

484. The interest equals the continued product of the 
principal., the rate per cent, expressed as hundredths, and the 
time expressed in years. If the interest is known, any one of 
the three other quantities may be found by dividing the inter- 
est of the product by the other two factors. 

WRITTEN PROBLEMS. 

485. 1. Find the interest with the following quantities 
given : 





Prin. 


Rate. 


Time. 




Prin. 


Rate. 


Time,. 


7. 


$750 


6% 


2 yrs. 8 mos. 


5. 


$846. 


ifc 


41 yrs. 


£. 


$525 


8% 


2 yrs. 9 mos. 


6. 


$398. 


% 


7 yrs. 


S. 


$475 


5% 


3 yrs. 4 mos. 


7. 


$652.50 


H 


3 yrs. 


4. 


$1000 


3% 


1 yr. 10 mos. 


8. 


$750.25 


±% 


44 yrs. 



2. What principal at 8% in 2 years will give $75 interest ? 

3. What rate will cause $450 in 3 years to earn $54 ? 
^. In how many years will $1200 earn $1152 at 8% ? 



INTEREST, 205 

486. To find the interest at 6fo for any number of 
days, 

487. If for 1 year the interest is 6% or T |^ of the princi- 
pal, for 1 day, it is 3^ part of T ^ ? or -^^^ or ^Vtf of the 
principal. 

488. Rule. — Divide the principal by 6000 and mul- 
tiply by the number of days ; or, multiply the principal 
by the number of days and divide by 6000. 

489. Rule for Finding the Amount. — Find the in- 
terest and add it to the principal. 

WRITTEN PROBLEMS. 

490. 1. Find the interest at 6% on : 

1. $750 for 46 da. 3. $315 for 29 da. 5. $210 for 33 da. 

2. $225 " 50 " 4. $678 " 63 " 6. $435 " 10 " 

2. Find the amount at 6% on : 

1. $220 for 20 da. 3. $1000 for 13 da. 5. $1250 for 30 da. 

2. $650 " 93 " 4: $2560 " 33 " e. $3000 " 60 " 

3. What is the interest on $697 for 3 yrs. 7 mos. 11 days at 
b% ? What is the amount ? 

4' What is the interest on $450 for 2 yrs. 11 mos. 5 days at 
3% ? What is the amount ? 

491. Rule. — The interest at any rate may be found, 
by first finding the interest at 6%, and then increasing 
or decreasing this interest by such part of itself as will 
0ive the interest at the required rate. 

2%=%of6%. ±%=6%-2%. b%=%%-\%. ltf c = iol6f c . 
3%=l-ol6f c . $%=6% + 2%. 7%=6% + l%. 4±%=6%-llfe. 

492. Find the interest on: 

1. $250 at 2% for 50 days. 4. $375 at b% for 33 days. 

2. $500 at 3% for 75 days. s. $150 at 7% for 63 days. 

3. $675 at \\% for 80 days. 6. $295 at 8% for 93 days. 



206 INTEREST. 

493. Banks advance money on promissory notes, secured 
by two responsible parties, deducting from the sum named in 
the note a discount, or interest on said sum, for the given 
time plus three days. The owner of the note receives the 
remainder, or proceeds in exchange for the note. 

494. Find the bank discount and the proceeds on the 
follotving : 

i. $630 @ 6%, 90 days. 5. $1000 @ 4$, 10 days. 

2. $475 @ 5%, 60 days. 6. $2500 @ 4^%, 20 days. 

3. $290 @ %, 30 days. 7. $1250 @ 9%, 10 days. 

4. $325 @ 9%, 45 days. *. $5000 @ b%, 50 days. 

495. A promissory note is a written promise to pay 
absolutely at a given time and place, and to the order of a 
certain person, a specified sum of money. 

496. The maker of the note is the person that makes the 
promise. 

497. The payee is the one to whom the promise is made. 

498. The holder is the owner of the note. 

499. The face of the note is the sum of money named 
in the note. 

500. The indorser is the one that writes his name on the 
back of the note, and thereby becomes security for its pay- 
ment. 

501. The bank discount is the sum charged by a bank 
as interest on the face of the note, for the time specified plus 
three days, for advancing the money before it is due. 

502. The days of grace are the three days allowed by 
custom for the payment of a note after it is due, and for 
which the bank charges discount. 

503. The proceeds is the sum paid by the bank to the 
holder of the note after deducting the bank discount. 



PROFIT AND LOSS. 207 

PROFIT AND LOSS. 

1. How much is .01 or 1$ of $400 ? How much is .10 or 
IO7; of $400 ? 

2. How much is 10$ of $500 ? Of $600 ? Of $900 ? 

3. How much is 10$ (= T V) of $1 ? Of $.50 ? Of $.40 ? 
Of $.10? 

^. Mr. Fenn bought muslin for $.10 a yard, and sold it 
so as to make 10$. How much did he make on a yard ? 
What was the selling price per yard ? 

5. A man bought a horse for $100 and sold it at a loss of 
20$ (=i). What was the selling price ? 

6. A merchant bought some buttons for $.25 per dozen 
and sold them at a loss of 20$. How much per dozen did 
he lose ? 

7. To gain 10$, what must be the selling price of goods 
costing $.10? $.20? $.40? $.70? $.90? 

8. To gain 5$ ( = ^), what must be the selling price of 
goods costing $.20 ? $.40 ? $1 ? 

9. To lose 12|-$ (=-§-)> what must be the selling price of 
goods costing $.08 ? $.16? $.24? $.40? $.72? $.80? 

WRITTEN PROBLEMS. 

504. 1. A man sold a carriage for $225, which was only 
75$ of the cost. What was 1$ of the cost ? What was the 
cost or 100$ ? 

2. If a horse costing $250 is sold for $300, how much is 
made by the sale ? $50 is what part of $250 ? One fifth 
equals how many hundredths ? How many per cent. ? 

3. A man paid 2$ to an agent for buying him a farm for 
$6000. What was the entire cost ? 

Jj,. After deducting 5$ for his services in the sale of a bill 
of goods, an agent returned 95$ or $1800 to the owner or 
principal. If 95$ equals $1800, how much is 1$ ? How 
much is 100$ or the selling price ? 



208 MEASUREMENT. 

MEASUREMENT. 

LENGTHS. 

505. 1. How Long Measure. 
long is your slate? M incheg (Jn } make t f 

How wide is it ? „ . . „ . . , 

« Make a line * feet 1 yard, ytf. 

^ Make a line 5* yds. or 16* ft. << 1 rod, rd 

6 in. long ; is it as 32Q rdg> Qr 528Q ft ^ „ x mil m{ 

wide as your slate r 

<?. How many feet wide is the school room ? How long is 
it ? How high ? 

Jf. How high is the school house ? How wide is the lot on 
which the school house is built ? How deep is it ? 

5. To what place will a mile from the school house extend ? 

6. How many inches are there in : 

1. 2 ft. ? 3. 2j ft. ? 5. l-i ft. ? ?. 1 yd. ? 

£. I ft. ? 4. i ft. ? <?. 1 yd. ? 8. 1 yd. ? 

7. What part of a foot are 6 in.? 4 in.? 3 in. ? 9 in. ? 8 in. ? 
5. What part of a yard is 1 ft. ? 2 ft. ? 1 J ft. ? 

SURFACES. 

506. 1. How many Square Measure. 

in. are there in 1 ft. ? 144 sq# in> make ± sq> ft# 
Make a line 1 ft. long ; 9 sq# f t . « t sq . y( j. 

divide it into inches. 30 £ sq- yds< « x sq> rd . 

2. How many sq. in. 160 gq# rds< « ± A ^ cre y 
are there in a sq. ft.? 640 acres " 1 sq. mi. 
Make a square figure with 

each of the four sides equal to 1 ft. Divide it into sq. in. 

3. How long is your teacher's desk ? Is the top of the 
desk 1 yd. square ? 

Jf. How many feet wide is the school room ? How many 
rods wide ? 

5. How large is the floor of the school room ? 1 sq. rd. ? 



MEASUREMENT. 



209 



50*7. A surface has only length, and breadth ; as, a piece 
of paper, a pane of glass. 

508. A rectangle is a surface bounded by four straight 
lines, and having four square or right angles for corners. 

509. A square is a rectangle with four equal sides. 

510. Tlie area of a rectangle is found by multiplying 
its length by its breadth. 

In measuring a surface to 
find its area or extent, some 
other surface, usually smaller 
and of square form, is used as 
a unit of measure. The area 
of a surface is then found by 
observing how many times it 
contains the unit of surface. 

Thus, if a rectangle is 4 in. long and 3 in. wide, it may be divided 
into 4x3, or 12 small squares, each of whose sides is 1 in. long. The 
square inch is then the unit of measure, and as the rectangle contains 
the square inch 12 times, its area is 12 sq. in., or the number of square 
inches found by multiplying its length by its breadth. 



■■■■■■■■■■ 

■■■■■■■■■a 
■■■«■■■■■■ 
■■■■■■■■■a 



WRITTEN PROBLEMS. 

511. i. Find the area of a floor 18 feet long, and 15 feet 
wide. How many square feet are there in the floor ? 

2. How many sq. ft. are there in a sidewalk 12 ft. wide, 
and 45 ft. long ? 

3. How wide is a blackboard, if its area is 45 sq. ft., and 
its length 15 ft. ? 

SOLIDS. 

512. A solid or volume has length, breadth, and thick- 
ness ; as, a box, a wall. 

513. A cube is a solid whose surface consists of six equal 
squares or faces. 



210 



MEASUREMENT. 



514. The volume or contents of a solid is the space it 
occupies. 

515. The volume of a solid whose angles are right 
angles is the product of its length, breadth, and thick- 
ness* 



WRITTEN PROBLEMS. 

516. 1. How many cu. Cubic Measure. 

ft. of earth were removed in 1728 cu. in. make 1 cu. ft. 

digging a cellar 8 ft. deep, 27 cu. ft. " 1 cu. yd. 

20 ft. wide, and 40 ft. long ? 16 cu. ft. " 1 cord ft. 

2. How many cords of 128 cu. ft. " 1 cord, 
wood are there in a pile 12 

ft. high, 8 ft. wide, and 20 ft. long ? 
at $6 a cord ? 

WEIGHTS. 

517. Troy weight is used in weighing gold, silver, and 
jewels ; and Apothecaries weight is used by druggists in 
weighing drugs not liquid. 



How much will it cost 



Apothecaries Weight. 
20 gr. make 1 scruple, 3. 

3 'sera. " 1 dram, 3 . 

8 drs. " 1 ounce, § . 
12 oz. " 1 pound, ft>. 

Note. — For the table of avoirdupois weights see page 121. Avoirdu- 
pois weight is used in buying and selling drugs at wholesale. 



Troy Weight. 
24: gr. make 1 pennyweight, pwt. 
20 pwt. " 1 ounce, oz. 

12 oz. " 1 pound, lb. or lb. 



MISCELLANEOUS TABLES. 

518. The following are the tables of denominations used 
in buying and selling paper and other things : 



12 units make 1 dozen. 



20 " 

12 dozen 
12 gross 



1 score. 
1 gross. 
1 great gross. 



24 sheets make 1 quire, qr. 



20 quires 
2 reams 

5 bundles 



1 ream, rm. 
1 bundle, bun. 
1 bale, B. 



RD -97 t 














k t • o. 






A <> ♦7R7* o* ^ •?•"• VV ^ ♦7ET» .6 1 







003 650 651 9 




